Greatest Common Divisor from Wikipedia, the Free Encyclopedia Contents

Greatest common divisor From Wikipedia, the free encyclopedia Contents

1 Augmented matrix 1 1.1 Examples ...... 1 1.1.1 Matrix inverse ...... 1 1.1.2 Existence and number of solutions ...... 2 1.1.3 Solution of a linear system ...... 2 1.2 References ...... 3

2 Coeﬃcient 4 2.1 Linear algebra ...... 4 2.2 Examples of physical coeﬃcients ...... 5 2.3 See also ...... 5 2.4 References ...... 5

3 Coeﬃcient matrix 6 3.1 Example ...... 6 3.2 See also ...... 6

4 Determinant 7 4.1 Deﬁnition ...... 7 4.1.1 2 × 2 matrices ...... 8 4.1.2 3 × 3 matrices ...... 10 4.1.3 n × n matrices ...... 10 4.2 Properties of the determinant ...... 12 4.2.1 Multiplicativity and matrix groups ...... 14 4.2.2 Laplace’s formula and the adjugate matrix ...... 14 4.2.3 Sylvester’s determinant theorem ...... 15 4.3 Properties of the determinant in relation to other notions ...... 15 4.3.1 Relation to eigenvalues and trace ...... 15 4.3.2 Cramer’s rule ...... 17 4.3.3 Block matrices ...... 17 4.3.4 Derivative ...... 18 4.4 Abstract algebraic aspects ...... 19 4.4.1 Determinant of an endomorphism ...... 19

i ii CONTENTS

4.4.2 Exterior algebra ...... 19 4.4.3 Square matrices over commutative rings and abstract properties ...... 20 4.5 Generalizations and related notions ...... 21 4.5.1 Inﬁnite matrices ...... 21 4.5.2 Related notions for non-commutative rings ...... 21 4.5.3 Further variants ...... 21 4.6 Calculation ...... 21 4.6.1 Decomposition methods ...... 21 4.6.2 Further methods ...... 22 4.7 History ...... 22 4.8 Applications ...... 23 4.8.1 Linear independence ...... 23 4.8.2 Orientation of a basis ...... 23 4.8.3 Volume and Jacobian determinant ...... 24 4.8.4 Vandermonde determinant (alternant) ...... 24 4.8.5 Circulants ...... 25 4.9 See also ...... 25 4.10 Notes ...... 25 4.11 References ...... 27 4.12 External links ...... 27

5 Greatest common divisor 28 5.1 Overview ...... 28 5.1.1 Notation ...... 28 5.1.2 Example ...... 28 5.1.3 Reducing fractions ...... 29 5.1.4 Coprime numbers ...... 29 5.1.5 A geometric view ...... 29 5.2 Calculation ...... 29 5.2.1 Using prime factorizations ...... 29 5.2.2 Using Euclid’s algorithm ...... 30 5.2.3 Binary method ...... 30 5.2.4 Other methods ...... 31 5.3 Properties ...... 32 5.4 Probabilities and expected value ...... 33 5.5 The gcd in commutative rings ...... 33 5.6 See also ...... 34 5.7 Notes ...... 34 5.8 References ...... 35 5.9 Further reading ...... 35 5.10 External links ...... 35 CONTENTS iii

6 Linear equation 38 6.1 One variable ...... 39 6.2 Two variables ...... 39 6.2.1 Forms for two-dimensional linear equations ...... 39 6.2.2 Connection with linear functions ...... 43 6.2.3 Examples ...... 44 6.3 More than two variables ...... 44 6.4 See also ...... 44 6.5 Notes ...... 44 6.6 References ...... 44 6.7 External links ...... 45

7 Mathematics 46 7.1 History ...... 47 7.1.1 Evolution ...... 47 7.1.2 Etymology ...... 49 7.2 Deﬁnitions of mathematics ...... 50 7.2.1 Mathematics as science ...... 50 7.3 Inspiration, pure and applied mathematics, and aesthetics ...... 53 7.4 Notation, language, and rigor ...... 54 7.5 Fields of mathematics ...... 54 7.5.1 Foundations and philosophy ...... 55 7.5.2 Pure mathematics ...... 56 7.5.3 Applied mathematics ...... 58 7.6 Mathematical awards ...... 58 7.7 See also ...... 59 7.8 Notes ...... 59 7.9 References ...... 61 7.10 Further reading ...... 62 7.11 External links ...... 62

8 Matrix (mathematics) 64 8.1 Deﬁnition ...... 65 8.1.1 Size ...... 65 8.2 Notation ...... 66 8.3 Basic operations ...... 66 8.3.1 Addition, scalar multiplication and transposition ...... 66 8.3.2 Matrix multiplication ...... 67 8.3.3 Row operations ...... 68 8.3.4 Submatrix ...... 68 8.4 Linear equations ...... 69 8.5 Linear transformations ...... 69 iv CONTENTS

8.6 Square matrices ...... 69 8.6.1 Main types ...... 70 8.6.2 Main operations ...... 72 8.7 Computational aspects ...... 73 8.8 Decomposition ...... 74 8.9 Abstract algebraic aspects and generalizations ...... 75 8.9.1 Matrices with more general entries ...... 75 8.9.2 Relationship to linear maps ...... 76 8.9.3 Matrix groups ...... 76 8.9.4 Inﬁnite matrices ...... 77 8.9.5 Empty matrices ...... 77 8.10 Applications ...... 78 8.10.1 Graph theory ...... 78 8.10.2 Analysis and geometry ...... 78 8.10.3 Probability theory and statistics ...... 79 8.10.4 Symmetries and transformations in physics ...... 80 8.10.5 Linear combinations of quantum states ...... 81 8.10.6 Normal modes ...... 81 8.10.7 Geometrical optics ...... 82 8.10.8 Electronics ...... 82 8.11 History ...... 82 8.11.1 Other historical usages of the word “matrix” in mathematics ...... 83 8.12 See also ...... 83 8.13 Notes ...... 84 8.14 References ...... 87 8.14.1 Physics references ...... 89 8.14.2 Historical references ...... 90 8.15 External links ...... 90

9 Numerical linear algebra 92 9.1 See also ...... 92 9.2 References ...... 92 9.3 External links ...... 93

10 System of linear equations 94 10.1 Elementary example ...... 95 10.2 General form ...... 95 10.2.1 Vector equation ...... 96 10.2.2 Matrix equation ...... 96 10.3 Solution set ...... 96 10.3.1 Geometric interpretation ...... 97 10.3.2 General behavior ...... 98 CONTENTS v

10.4 Properties ...... 99 10.4.1 Independence ...... 99 10.4.2 Consistency ...... 99 10.4.3 Equivalence ...... 101 10.5 Solving a linear system ...... 101 10.5.1 Describing the solution ...... 102 10.5.2 Elimination of variables ...... 102 10.5.3 Row reduction ...... 103 10.5.4 Cramer’s rule ...... 103 10.5.5 Matrix solution ...... 104 10.5.6 Other methods ...... 104 10.6 Homogeneous systems ...... 105 10.6.1 Solution set ...... 105 10.6.2 Relation to nonhomogeneous systems ...... 105 10.7 See also ...... 106 10.8 Notes ...... 106 10.9 References ...... 106 10.9.1 Textbooks ...... 106

11 Vector space 107 11.1 Introduction and definition ...... 108 11.1.1 First example: arrows in the plane ...... 108 11.1.2 Second example: ordered pairs of numbers ...... 108 11.1.3 Definition ...... 108 11.1.4 Alternative formulations and elementary consequences ...... 109 11.2 History ...... 109 11.3 Examples ...... 110 11.3.1 Coordinate spaces ...... 110 11.3.2 Complex numbers and other field extensions ...... 110 11.3.3 Function spaces ...... 110 11.3.4 Linear equations ...... 111 11.4 Basis and dimension ...... 111 11.5 Linear maps and matrices ...... 113 11.5.1 Matrices ...... 114 11.5.2 Eigenvalues and eigenvectors ...... 115 11.6 Basic constructions ...... 116 11.6.1 Subspaces and quotient spaces ...... 116 11.6.2 Direct product and direct sum ...... 117 11.6.3 Tensor product ...... 117 11.7 Vector spaces with additional structure ...... 118 11.7.1 Normed vector spaces and inner product spaces ...... 118 11.7.2 Topological vector spaces ...... 119 vi CONTENTS

11.7.3 Algebras over fields ...... 122 11.8 Applications ...... 123 11.8.1 Distributions ...... 123 11.8.2 Fourier analysis ...... 124 11.8.3 Differential geometry ...... 125 11.9 Generalizations ...... 126 11.9.1 Vector bundles ...... 126 11.9.2 Modules ...... 127 11.9.3 Affine and projective spaces ...... 127 11.10See also ...... 127 11.11Notes ...... 127 11.12Footnotes ...... 129 11.13References ...... 132 11.13.1 Algebra ...... 132 11.13.2 Analysis ...... 132 11.13.3 Historical references ...... 133 11.13.4 Further references ...... 133 11.14External links ...... 135 11.15Text and image sources, contributors, and licenses ...... 136 11.15.1 Text ...... 136 11.15.2 Images ...... 141 11.15.3 Content license ...... 145 Chapter 1

Augmented matrix

In linear algebra, an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices. Given the matrices A and B, where     1 3 2 4 A = 2 0 1,B = 3, 5 2 2 1 the augmented matrix (A|B) is written as   1 3 2 4 (A|B) =  2 0 1 3  . 5 2 2 1 This is useful when solving systems of linear equations. For a given number of unknowns, the number of solutions to a system of linear equations depends only on the rank of the matrix representing the system and the rank of the corresponding augmented matrix. Specifically, according to the Rouché–Capelli theorem, any system of linear equations is inconsistent (has no solutions) if the rank of the augmented matrix is greater than the rank of the coefficient matrix; if, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has k free parameters where k is the difference between the number of variables and the rank; hence in such a case there are an infinitude of solutions. An augmented matrix may also be used to find the inverse of a matrix by combining it with the identity matrix.

1.1 Examples

1.1.1 Matrix inverse

Let C be the square 2×2 matrix

[ ] 1 3 C = . −5 0 To ﬁnd the inverse of C we create (C|I) where I is the 2×2 identity matrix. We then reduce the part of (C|I) corresponding to C to the identity matrix using only elementary row operations on (C|I).

[ ] 1 3 1 0 (C|I) = −5 0 0 1 [ ] − 1 −1 1 0 0 5 (I|C ) = 1 1 0 1 3 15

1 2 CHAPTER 1. AUGMENTED MATRIX the right part of which is the inverse of the original matrix.

1.1.2 Existence and number of solutions

Consider the system of equations

x + y + 2z = 3 x + y + z = 1

2x + 2y + 2z = 2.

The coeﬃcient matrix is

  1 1 2 A = 1 1 1, 2 2 2 and the augmented matrix is

  1 1 2 3 (A|B) =  1 1 1 1  . 2 2 2 2

Since both of these have the same rank, namely 2, there exists at least one solution; and since their rank is less than the number of unknowns, the latter being 3, there are an inﬁnite number of solutions. In contrast, consider the system

x + y + 2z = 3 x + y + z = 1 2x + 2y + 2z = 5.

The coeﬃcient matrix is

  1 1 2 A = 1 1 1, 2 2 2 and the augmented matrix is

  1 1 2 3 (A|B) =  1 1 1 1  . 2 2 2 5

In this example the coeﬃcient matrix has rank 2 while the augmented matrix has rank 3; so this system of equations has no solution. Indeed, an increase in the number of linearly independent rows has made the system of equations inconsistent.

1.1.3 Solution of a linear system

As used in linear algebra, an augmented matrix is used to represent the coeﬃcients and the solution vector of each equation set. For the set of equations 1.2. REFERENCES 3

x + 2y + 3z = 0 3x + 4y + 7z = 2 6x + 5y + 9z = 11 the coeﬃcients and constant terms give the matrices

    1 2 3 0 A = 3 4 7,B =  2 , 6 5 9 11 and hence give the augmented matrix

  1 2 3 0 (A|B) =  3 4 7 2  6 5 9 11

Note that the rank of the coeﬃcient matrix, which is 3, equals the rank of the augmented matrix, so at least one solution exists; and since this rank equals the number of unknowns, there is exactly one solution. To obtain the solution, row operations can be performed on the augmented matrix to obtain the identity matrix on the left side, yielding

  1 0 0 4  0 1 0 1  , 0 0 1 −2 so the solution of the system is (x, y, z) = (4, 1, −2).

1.2 References

• Marvin Marcus and Henryk Minc, A survey of matrix theory and matrix inequalities, Dover Publications, 1992, ISBN 0-486-67102-X. Page 31. Chapter 2

Coeﬃcient

For other uses, see Coeﬃcient (disambiguation).

In mathematics, a coeﬃcient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but in any case does not involve any variables of the expression. For instance in

7x2 − 3xy + 1.5 + y the first two terms respectively have the coefficients 7 and −3. The third term 1.5 is a constant. The final term does not have any explicitly written coefficient, but is considered to have coefficient 1, since multiplying by that factor would not change the term. Often coefficients are numbers as in this example, although they could be parameters of the problem, as a, b, and c, where “c” is a constant, in

ax2 + bx + c when it is understood that these are not considered variables. Thus a polynomial in one variable x can be written as

k 1 akx + ··· + a1x + a0

for some integer k , where ak, . . . , a1, a0 are coefficients; to allow this kind of expression in all cases one must allow introducing terms with 0 as coefficient. For the largest i with ai ≠ 0 (if any), ai is called the leading coefficient of the polynomial. So for example the leading coefficient of the polynomial

4x5 + x3 + 2x2 is 4. Specific coefficients arise in mathematical identities, such as the binomial theorem which involves binomial coefficients; these particular coefficients are tabulated in Pascal’s triangle.

2.1 Linear algebra

In linear algebra, the leading coeﬃcient of a row in a matrix is the ﬁrst nonzero entry in that row. So, for example, given

  1 2 0 6 0 2 9 4 M =   0 0 0 4 0 0 0 0

4 2.2. EXAMPLES OF PHYSICAL COEFFICIENTS 5

The leading coefficient of the first row is 1; 2 is the leading coefficient of the second row; 4 is the leading coefficient of the third row, and the last row does not have a leading coefficient. Though coefficients are frequently viewed as constants in elementary algebra, they can be variables more generally. For example, the coordinates (x1, x2, . . . , xn) of a vector v in a vector space with basis {e1, e2, . . . , en} , are the coefficients of the basis vectors in the expression

v = x1e1 + x2e2 + ··· + xnen.

2.2 Examples of physical coeﬃcients

1. Coefficient of Thermal Expansion (thermodynamics) (dimensionless) - Relates the change in temperature to the change in a material’s dimensions. 2. Partition Coefficient (KD)(chemistry) - The ratio of concentrations of a compound in two phases of a mixture of two immiscible solvents at equilibrium. H2O is a coefficient 3. Hall coefficient (electrical physics) - Relates a magnetic field applied to an element to the voltage created, the amount of current and the element thickness. It is a characteristic of the material from which the conductor is made. 4. Lift coefficient (CL or CZ)(Aerodynamics) (dimensionless) - Relates the lift generated by an airfoil with the dynamic pressure of the fluid flow around the airfoil, and the plan-form area of the airfoil. 5. Ballistic coefficient (BC) (Aerodynamics) (units of kg/m2) - A measure of a body’s ability to overcome air resistance in flight. BC is a function of mass, diameter, and drag coefficient. 6. Transmission Coefficient (quantum mechanics) (dimensionless) - Represents the probability flux of a transmitted wave relative to that of an incident wave. It is often used to describe the probability of a particle tunnelling through a barrier. 7. Damping Factor a.k.a. viscous damping coefficient (Physical Engineering) (units of newton-seconds per meter) - relates a damping force with the velocity of the object whose motion is being damped.

A coeﬃcient is a number placed in front of a term in a chemical equation to indicate how many molecules (or atoms) take part in the reaction. For example, in the formula

2H2 + O2 → 2H2O the number 2’s in front of H2 and H2O are stoichiometric coeﬃcients.

2.3 See also

• Degree of a polynomial • Monic polynomial

2.4 References

• Sabah Al-hadad and C.H. Scott (1979) College Algebra with Applications, page 42, Winthrop Publishers, Cam- bridge Massachusetts ISBN 0-87626-140-3 . • Gordon Fuller, Walter L Wilson, Henry C Miller, (1982) College Algebra, 5th edition, page 24, Brooks/Cole Publishing, Monterey California ISBN 0-534-01138-1 . • Steven Schwartzman (1994) The Words of Mathematics: an etymological dictionary of mathematical terms used in English, page 48, Mathematics Association of America, ISBN 0-88385-511-9. Chapter 3

Coeﬃcient matrix

In linear algebra, the coeﬃcient matrix refers to a matrix consisting of the coeﬃcients of the variables in a set of linear equations.

3.1 Example

In general, a system with m linear equations and n unknowns can be written as

a11x1 + a12x2 + ··· + a1nxn = b1 a21x1 + a22x2 + ··· + a2nxn = b2 . . am1x1 + am2x2 + ··· + amnxn = bm

where x1, x2, ..., xn are the unknowns and the numbers a11, a12, ..., amn are the coefficients of the system. The coefficient matrix is the mxn matrix with the coefficient aij as the (i,j)-th entry:

  a11 a12 ··· a1n    a21 a22 ··· a2n   . . . .   . . .. .  am1 am2 ··· amn

3.2 See also

• System of linear equations

6 Chapter 4

Determinant

This article is about determinants in mathematics. For determinants in epidemiology, see risk factor.

In linear algebra, the determinant is a useful value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. In the case of a 2 × 2 matrix, the speciﬁc formula for the determinant is simply the upper left element times the lower right element, minus the product of the other two elements. Similarly, suppose we have a 3 × 3 matrix A, and we want the speciﬁc formula for its determinant |A|:

a b c e f d f d e |A| = d e f = a − b + c = aei + bfg + cdh − ceg − bdi − afh. h i g i g h g h i

Each determinant of a 2 × 2 matrix in this equation is called a "minor" of the matrix A. The same sort of procedure can be used to find the determinant of a 4 × 4 matrix, the determinant of a 5 × 5 matrix, and so forth. Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and the determinant is used to solve those equations. The use of determinants in calculus includes the Jacobian determinant in the change of variables rule for integrals of functions of several variables. Determinants are also used to define the characteristic polynomial of a matrix, which is essential for eigenvalue problems in linear algebra. Sometimes, determinants are used merely as a compact notation for expressions that would otherwise be unwieldy to write down. It can be proven that any matrix has a unique inverse if its determinant is nonzero. Various other theorems can be proved as well, including that the determinant of a product of matrices is always equal to the product of determinants; and, the determinant of a Hermitian matrix is always real.

4.1 Deﬁnition

There are various ways to deﬁne the determinant of a square matrix A, i.e. one with the same number of rows and columns. Perhaps the simplest way to express the determinant is by considering the elements in the top row and the respective minors; starting at the left, multiply the element by the minor, then subtract the product of the next element and its minor, and alternate adding and subtracting such products until all elements in the top row have been exhausted. For example, here is the result for a 4 × 4 matrix:

a b c d f g h e g h e f h e f g e f g h = a j k l −b i k l +c i j l −d i j k . i j k l n o p m o p m n p m n o m n o p

Another way to deﬁne the determinant is expressed in terms of the columns of the matrix. If we write an n × n matrix A in terms of its column vectors

7 8 CHAPTER 4. DETERMINANT

[ ] A = a1, a2, . . . , an

where the aj are vectors of size n, then the determinant of A is deﬁned so that

[ ] [ ] det a , . . . , ba + cv, . . . , a = b det(A) + c det a , ..., v, ...,a [ 1 j n ] [ 1 n ] det a1, . . . , aj, aj+1, . . . , an = − det a1, . . . , aj+1, aj, . . . , an det(I) = 1

where b and c are scalars, v is any vector of size n and I is the identity matrix of size n. These equations say that the determinant is a linear function of each column, that interchanging adjacent columns reverses the sign of the determinant, and that the determinant of the identity matrix is 1. These properties mean that the determinant is an alternating multilinear function of the columns that maps the identity matrix to the underlying unit scalar. These suffice to uniquely calculate the determinant of any square matrix. Provided the underlying scalars form a field (more generally, a commutative ring with unity), the definition below shows that such a function exists, and it can be shown to be unique.[1] Equivalently, the determinant can be expressed as a sum of products of entries of the matrix where each product has n terms and the coefficient of each product is −1 or 1 or 0 according to a given rule: it is a polynomial expression of the matrix entries. This expression grows rapidly with the size of the matrix (an n × n matrix contributes n! terms), so it will first be given explicitly for the case of 2 × 2 matrices and 3 × 3 matrices, followed by the rule for arbitrary size matrices, which subsumes these two cases. Assume A is a square matrix with n rows and n columns, so that it can be written as

  a1,1 a1,2 . . . a1,n   a2,1 a2,2 . . . a2,n  A =  . . . . .  . . .. .  an,1 an,2 . . . an,n

The entries can be numbers or expressions (as happens when the determinant is used to deﬁne a characteristic polynomial); the deﬁnition of the determinant depends only on the fact that they can be added and multiplied together in a commutative manner. The determinant of A is denoted as det(A), or it can be denoted directly in terms of the matrix entries by writing enclosing bars instead of brackets:

a1,1 a1,2 . . . a1,n

a2,1 a2,2 . . . a2,n ......

an,1 an,2 . . . an,n

4.1.1 2 × 2 matrices

The determinant of a 2 × 2 matrix is deﬁned by

a b = ad − bc. c d

If the matrix entries are real numbers, the matrix A can be used to represent two linear maps: one that maps the standard basis vectors to the rows of A, and one that maps them to the columns of A. In either case, the images of the basis vectors form a parallelogram that represents the image of the unit square under the mapping. The parallelogram deﬁned by the rows of the above matrix is the one with vertices at (0, 0), (a, b), (a + c, b + d), and (c, d), as shown in the accompanying diagram. 4.1. DEFINITION 9

(a+c,b+d)

(c,d)

ad−bc

(a,b)

(0,0)

The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram’s sides.

The absolute value of ad − bc is the area of the parallelogram, and thus represents the scale factor by which areas are transformed by A. (The parallelogram formed by the columns of A is in general a different parallelogram, but since the determinant is symmetric with respect to rows and columns, the area will be the same.) The absolute value of the determinant together with the sign becomes the oriented area of the parallelogram. The oriented area is the same as the usual area, except that it is negative when the angle from the first to the second vector defining the parallelogram turns in a clockwise direction (which is opposite to the direction one would get for the identity matrix). Thus the determinant gives the scaling factor and the orientation induced by the mapping represented by A. When the determinant is equal to one, the linear mapping defined by the matrix is equi-areal and orientation-preserving. The object known as the bivector is related to these ideas. In 2D, it can be interpreted as an oriented plane segment formed by imagining two vectors each with origin (0, 0), and coordinates (a, b) and (c, d). The bivector magnitude 10 CHAPTER 4. DETERMINANT

(denoted (a, b) ∧ (c, d)) is the signed area, which is also the determinant ad − bc.[2]

4.1.2 3 × 3 matrices

The volume of this parallelepiped is the absolute value of the determinant of the matrix formed by the rows constructed from the vectors r1, r2, and r3.

The determinant of a 3 × 3 matrix is deﬁned by

a b c e f d f d e d e f = a − b + c h i g i g h g h i = a(ei − fh) − b(di − fg) + c(dh − eg) = aei + bfg + cdh − ceg − bdi − afh.

The rule of Sarrus is a mnemonic for the 3 × 3 matrix determinant: the sum of the products of three diagonal north- west to south-east lines of matrix elements, minus the sum of the products of three diagonal south-west to north-east lines of elements, when the copies of the ﬁrst two columns of the matrix are written beside it as in the illustration. This scheme for calculating the determinant of a 3 × 3 matrix does not carry over into higher dimensions.

4.1.3 n × n matrices

The determinant of a matrix of arbitrary size can be deﬁned by the Leibniz formula or the Laplace formula. The Leibniz formula for the determinant of an n × n matrix A is 4.1. DEFINITION 11

Sarrus’ rule: The determinant of the three columns on the left is the sum of the products along the solid diagonals minus the sum of the products along the dashed diagonals

∑ ∏n

det(A) = sgn(σ) ai,σi .

σ∈Sn i=1 Here the sum is computed over all permutations σ of the set {1, 2, ..., n}. A permutation is a function that reorders this set of integers. The value in the ith position after the reordering σ is denoted σi. For example, for n = 3, the original sequence 1, 2, 3 might be reordered to σ = [2, 3, 1], with σ1 = 2, σ2 = 3, and σ3 = 1. The set of all such permutations (also known as the symmetric group on n elements) is denoted Sn. For each permutation σ, sgn(σ) denotes the signature of σ, a value that is +1 whenever the reordering given by σ can be achieved by successively interchanging two entries an even number of times, and −1 whenever it can be achieved by an odd number of such interchanges. In any of the n! summands, the term

∏n

ai,σi i=1 is notation for the product of the entries at positions (i, σi), where i ranges from 1 to n:

· ··· a1,σ1 a2,σ2 an,σn . For example, the determinant of a 3 × 3 matrix A (n = 3) is

∑ ∏n ∏n ∏n ∏n

sgn(σ) ai,σi = sgn([1, 2, 3]) ai,[1,2,3]i + sgn([1, 3, 2]) ai,[1,3,2]i + sgn([2, 1, 3]) ai,[2,1,3]i

σ∈Sn i=1 i=1 i=1 i=1 ∏n ∏n ∏n

+ sgn([2, 3, 1]) ai,[2,3,1]i + sgn([3, 1, 2]) ai,[3,1,2]i + sgn([3, 2, 1]) ai,[3,2,1]i i=1 i=1 i=1 ∏n ∏n ∏n ∏n ∏n ∏n − − − = ai,[1,2,3]i ai,[1,3,2]i ai,[2,1,3]i + ai,[2,3,1]i + ai,[3,1,2]i ai,[3,2,1]i i=1 i=1 i=1 i=1 i=1 i=1

= a1,1a2,2a3,3 − a1,1a2,3a3,2 − a1,2a2,1a3,3 + a1,2a2,3a3,1

+ a1,3a2,1a3,2 − a1,3a2,2a3,1. 12 CHAPTER 4. DETERMINANT

Levi-Civita symbol

It is sometimes useful to extend the Leibniz formula to a summation in which not only permutations, but all sequences of n indices in the range 1, ..., n occur, ensuring that the contribution of a sequence will be zero unless it denotes a

permutation. Thus the totally antisymmetric Levi-Civita symbol εi1,··· ,in extends the signature of a permutation, by setting εσ(1),··· ,σ(n) = sgn(σ) for any permutation σ of n, and εi1,··· ,in = 0 when no permutation σ exists such that σ(j) = ij for j = 1, . . . , n (or equivalently, whenever some pair of indices are equal). The determinant for an n × n matrix can then be expressed using an n-fold summation as

∑n ··· det(A) = εi1···in a1,i1 an,in ,

i1,i2,...,in=1 or using two epsilon symbols as

1 ∑ det(A) = ε ··· ε ··· a ··· a , n! i1 in j1 jn i1j1 injn where now each ir and each jr should be summed over 1, ..., n.

4.2 Properties of the determinant

The determinant has many properties. Some basic properties of determinants are

1. det(In) = 1 where In is the n × n identity matrix. 2. det(AT) = det(A).

−1 1 −1 3. det(A ) = det(A) = det(A) . 4. For square matrices A and B of equal size,

det(AB) = det(A) det(B).

1. det(cA) = cn det(A) for an n × n matrix. 2. If A is a triangular matrix, i.e. ai,j = 0 whenever i > j or, alternatively, whenever i < j, then its determinant equals the product of the diagonal entries:

∏n det(A) = a1,1a2,2 ··· an,n = ai,i. i=1

This can be deduced from some of the properties below, but it follows most easily directly from the Leibniz formula (or from the Laplace expansion), in which the identity permutation is the only one that gives a non-zero contribution. A number of additional properties relate to the eﬀects on the determinant of changing particular rows or columns:

1. Viewing an n × n matrix as being composed of n columns, the determinant is an n-linear function. This means that if one column of a matrix A is written as a sum v + w of two column vectors, and all other columns are left unchanged, then the determinant of A is the sum of the determinants of the matrices obtained from A by replacing the column by v and then by w (and a similar relation holds when writing a column as a scalar multiple of a column vector). 2. If in a matrix, any row or column is 0, then the determinant of that particular matrix is 0. 4.2. PROPERTIES OF THE DETERMINANT 13

3. This n-linear function is an alternating form. This means that whenever two columns of a matrix are identical, or more generally some column can be expressed as a linear combination of the other columns (i.e. the columns of the matrix form a linearly dependent set), its determinant is 0.

Properties 1, 7 and 9 — which all follow from the Leibniz formula — completely characterize the determinant; in other words the determinant is the unique function from n × n matrices to scalars that is n-linear alternating in the columns, and takes the value 1 for the identity matrix (this characterization holds even if scalars are taken in any given commutative ring). To see this it suffices to expand the determinant by multi-linearity in the columns into a (huge) linear combination of determinants of matrices in which each column is a standard basis vector. These determinants are either 0 (by property 8) or else ±1 (by properties 1 and 11 below), so the linear combination gives the expression above in terms of the Levi-Civita symbol. While less technical in appearance, this characterization cannot entirely replace the Leibniz formula in defining the determinant, since without it the existence of an appropriate function is not clear. For matrices over non-commutative rings, properties 7 and 8 are incompatible for n ≥ 2,[3] so there is no good definition of the determinant in this setting. Property 2 above implies that properties for columns have their counterparts in terms of rows:

1. Viewing an n × n matrix as being composed of n rows, the determinant is an n-linear function.

2. This n-linear function is an alternating form: whenever two rows of a matrix are identical, its determinant is 0.

3. Interchanging any pair of columns or rows of a matrix multiplies its determinant by −1. This follows from properties 7 and 9 (it is a general property of multilinear alternating maps). More generally, any permutation of the rows or columns multiplies the determinant by the sign of the permutation. By permutation, it is meant viewing each row as a vector Ri (equivalently each column as Ci) and reordering the rows (or columns) by interchange of Rj and Rk (or Cj and Ck), where j,k are two indices chosen from 1 to n for an n × n square matrix.

4. Adding a scalar multiple of one column to another column does not change the value of the determinant. This is a consequence of properties 7 and 8: by property 7 the determinant changes by a multiple of the determinant of a matrix with two equal columns, which determinant is 0 by property 8. Similarly, adding a scalar multiple of one row to another row leaves the determinant unchanged.

Property 5 says that the determinant on n × n matrices is homogeneous of degree n. These properties can be used to facilitate the computation of determinants by simplifying the matrix to the point where the determinant can be determined immediately. Specifically, for matrices with coefficients in a field, properties 11 and 12 can be used to transform any matrix into a triangular matrix, whose determinant is given by property 6; this is essentially the method of Gaussian elimination. For example, the determinant of

  −2 2 −3 A = −1 1 3  2 0 −1 can be computed using the following matrices:

      −2 2 −3 −2 2 −3 −2 2 −3 B =  0 0 4.5,C =  0 0 4.5,D =  0 2 −4. 2 0 −1 0 2 −4 0 0 4.5

Here, B is obtained from A by adding −1/2×the ﬁrst row to the second, so that det(A) = det(B). C is obtained from B by adding the ﬁrst to the third row, so that det(C) = det(B). Finally, D is obtained from C by exchanging the second and third row, so that det(D) = −det(C). The determinant of the (upper) triangular matrix D is the product of its entries on the main diagonal: (−2) · 2 · 4.5 = −18. Therefore, det(A) = −det(D) = +18. 14 CHAPTER 4. DETERMINANT

4.2.1 Multiplicativity and matrix groups

The determinant of a matrix product of square matrices equals the product of their determinants:

det(AB) = det(A) det(B).

Thus the determinant is a multiplicative map. This property is a consequence of the characterization given above of the determinant as the unique n-linear alternating function of the columns with value 1 on the identity matrix, since the function Mn(K) → K that maps M ↦ det(AM) can easily be seen to be n-linear and alternating in the columns of M, and takes the value det(A) at the identity. The formula can be generalized to (square) products of rectangular matrices, giving the Cauchy–Binet formula, which also provides an independent proof of the multiplicative property. The determinant det(A) of a matrix A is non-zero if and only if A is invertible or, yet another equivalent statement, if its rank equals the size of the matrix. If so, the determinant of the inverse matrix is given by

1 det(A−1) = . det(A)

In particular, products and inverses of matrices with determinant one still have this property. Thus, the set of such matrices (of ﬁxed size n) form a group known as the special linear group. More generally, the word “special” indicates the subgroup of another matrix group of matrices of determinant one. Examples include the special orthogonal group (which if n is 2 or 3 consists of all rotation matrices), and the special unitary group.

4.2.2 Laplace’s formula and the adjugate matrix

Laplace’s formula expresses the determinant of a matrix in terms of its minors. The minor Mi,j is deﬁned to be the determinant of the (n−1) × (n−1)-matrix that results from A by removing the ith row and the jth column. The expression (−1)i+jMi,j is known as cofactor. The determinant of A is given by

∑n ∑n i+j i+j det(A) = (−1) ai,jMi,j = (−1) ai,jMi,j. j=1 i=1

Calculating det(A) by means of that formula is referred to as expanding the determinant along a row or column. For the example 3 × 3 matrix

  −2 2 −3 A = −1 1 3  , 2 0 −1

Laplace expansion along the second column (j = 2, the sum runs over i) yields: However, Laplace expansion is eﬃcient for small matrices only. The adjugate matrix adj(A) is the transpose of the matrix consisting of the cofactors, i.e.,

i+j (adj(A))i,j = (−1) Mj,i.

In terms of the adjugate matrix, Laplace’s expansion can be written as[4]

(det A)I = A adj A = (adj A) A. 4.3. PROPERTIES OF THE DETERMINANT IN RELATION TO OTHER NOTIONS 15

4.2.3 Sylvester’s determinant theorem

Sylvester’s determinant theorem states that for A, an m × n matrix, and B, an n × m matrix (so that A and B have dimensions allowing them to be multiplied in either order):

det(Im + AB) = det(In + BA) where Im and In are the m × m and n × n identity matrices, respectively. From this general result several consequences follow.

(a) For the case of column vector c and row vector r, each with m components, the formula allows quick calculation of the determinant of a matrix that diﬀers from the identity matrix by a matrix of rank 1:

det(Im + cr) = 1 + rc

(b) More generally,[5] for any invertible m × m matrix X,

−1 det(X + AB) = det(X) det(In + BX A) det(X + cr) = det(X)(1 + rX−1c) = det(X) + r adj(X) c

4.3 Properties of the determinant in relation to other notions

4.3.1 Relation to eigenvalues and trace

Main article: Eigenvalues and eigenvectors

Let A be an arbitrary n × n matrix of complex numbers with eigenvalues λ1 , λ2 , ... λn . (Here it is understood that an eigenvalue with algebraic multiplicities µ occurs µ times in this list.) Then the determinant of A is the product of all eigenvalues:

∏n det(A) = λi = λ1λ2 ··· λn i=1 The product of all non-zero eigenvalues is referred to as pseudo-determinant. Conversely, determinants can be used to ﬁnd the eigenvalues of the matrix A: they are the solutions of the characteristic equation det(A − xI) = 0, where I is the identity matrix of the same dimension as A. An Hermitian matrix is positive deﬁnite if all its eigenvalues are positive. Sylvester’s criterion asserts that this is equivalent to the determinants of the submatrices

  a1,1 a1,2 . . . a1,k   a2,1 a2,2 . . . a2,k Ak :=  . . . .   . . .. .  ak,1 ak,2 . . . ak,k 16 CHAPTER 4. DETERMINANT

being positive, for all k between 1 and n. The trace tr(A) is by deﬁnition the sum of the diagonal entries of A and also equals the sum of the eigenvalues. Thus, for complex matrices A,

det(exp(A)) = exp(tr(A)) or, for real matrices A,

tr(A) = log(det(exp(A))). Here exp(A) denotes the matrix exponential of A, because every eigenvalue λ of A corresponds to the eigenvalue exp(λ) of exp(A). In particular, given any logarithm of A, that is, any matrix L satisfying

exp(L) = A the determinant of A is given by

det(A) = exp(tr(L)). For example, for n = 2, n = 3, and n = 4, respectively,

( ) det(A) = (tr A)2 − tr(A2) /2, ( ) det(A) = (tr A)3 − 3 tr A tr(A2) + 2 tr(A3) /6, ( ) det(A) = (tr A)4 − 6 tr(A2)(tr A)2 + 3(tr(A2))2 + 8 tr(A3) tr A − 6 tr(A4) /24. cf. Cayley-Hamilton theorem. Such expressions are deducible from Newton’s identities. In the general case,[6]

∑ ∏n (−1)kl+1 l kl det(A) = k tr(A ) , l l kl! k1,k2,...,kn l=1 where the sum is taken over the set of all integers kl ≥ 0 satisfying the equation

∑n lkl = n. l=1 I This formula can also be used to ﬁnd the determinant of a matrix A J with multidimensional indices I = (i1,i2,...,iᵣ) and J = (j1,j2,...,jᵣ). The product and trace of such matrices are deﬁned in a natural way as

∑ ∑ I I K I (AB)J = AK BJ , tr(A) = AI . K I An arbitrary dimension n identity can be obtained from the Mercator series expansion of the logarithm when A ∈ B(0, 1)

 k ∞ ∞ ∑ 1 ∑ (−1)j det(I + A) = − tr(Aj) , k! j k=0 j=1 where I is the identity matrix. The sum and the expansion of the exponential only need to go up to n instead of ∞, since the determinant cannot exceed O(An). 4.3. PROPERTIES OF THE DETERMINANT IN RELATION TO OTHER NOTIONS 17

Upper and lower bounds

For a positive deﬁnite matrix A, the trace operator gives the following tight lower and upper bounds on the log determinant tr(I − A−1) ≤ log det(A) ≤ tr(A − I) with equality if and only if A = I . This relationship can be derived via the formula for the KL-divergence between two multivariate normal distributions.

4.3.2 Cramer’s rule

For a matrix equation

Ax = b the solution is given by Cramer’s rule:

det(A ) x = i i = 1, 2, 3, . . . , n i det(A) where Ai is the matrix formed by replacing the ith column of A by the column vector b. This follows immediately by column expansion of the determinant, i.e.

[ ] ∑n [ ] det(Ai) = det a1, . . . , b, . . . , an = xj det a1, . . . , ai−1, aj, ai+1, . . . , an = xi det(A) j=1 where the vectors aj are the columns of A. The rule is also implied by the identity

A adj(A) = adj(A) A = det(A) In.

It has recently been shown that Cramer’s rule can be implemented in O(n3) time,[7] which is comparable to more common methods of solving systems of linear equations, such as LU, QR, or singular value decomposition.

4.3.3 Block matrices

Suppose A, B, C, and D are matrices of dimension n × n, n × m, m × n, and m × m, respectively. Then

( ) ( ) A 0 AB det = det(A) det(D) = det . CD 0 D

This can be seen from the Leibniz formula, or from a decomposition like (for the former case)

( ) ( )( ) A 0 A 0 I 0 = n . CD CIm 0 D When A is invertible, one has

( ) AB det = det(A) det(D − CA−1B). CD 18 CHAPTER 4. DETERMINANT

as can be seen by employing the decomposition

( ) ( )( ) −1 AB A 0 In A B = −1 . CD CIm 0 D − CA B

When D is invertible, a similar identity with det(D) factored out can be derived analogously,[8] that is,

( ) AB det = det(D) det(A − BD−1C). CD

When the blocks are square matrices of the same order further formulas hold. For example, if C and D commute (i.e., CD = DC), then the following formula comparable to the determinant of a 2 × 2 matrix holds:[9]

( ) AB det = det(AD − BC). CD

When A = D and B = C, the blocks are square matrices of the same order and the following formula holds (even if A and B do not commute)

( ) AB det = det(A − B) det(A + B). BA

When D is a 1×1 matrix, B is a column vector, and C is a row vector then

( ) AB det = (D − 1) det(A) + det(A − BC) = (D + 1) det A − det(A + BC) . CD

4.3.4 Derivative

By deﬁnition, e.g., using the Leibniz formula, the determinant of real (or analogously for complex) square matrices is a polynomial function from Rn × n to R. As such it is everywhere diﬀerentiable. Its derivative can be expressed using Jacobi’s formula:[10]

( ) d det(A) dA = tr adj(A) . dα dα where adj(A) denotes the adjugate of A. In particular, if A is invertible, we have

( ) d det(A) dA = det(A) tr A−1 . dα dα Expressed in terms of the entries of A, these are

∂ det(A) −1 = adj(A)ji = det(A)(A )ji. ∂Aij Yet another equivalent formulation is det(A + ϵX) − det(A) = tr(adj(A)X)ϵ + O(ϵ2) = det(A) tr(A−1X)ϵ + O(ϵ2)

using big O notation. The special case where A = I , the identity matrix, yields 4.4. ABSTRACT ALGEBRAIC ASPECTS 19

det(I + ϵX) = 1 + tr(X)ϵ + O(ϵ2).

This identity is used in describing the tangent space of certain matrix Lie groups. [ ] If the matrix A is written as A = a b c where a, b, c are vectors, then the gradient over one of the three vectors may be written as the cross product of the other two:

∇a det(A) = b × c

∇b det(A) = c × a

∇c det(A) = a × b.

4.4 Abstract algebraic aspects

4.4.1 Determinant of an endomorphism

The above identities concerning the determinant of products and inverses of matrices imply that similar matrices have the same determinant: two matrices A and B are similar, if there exists an invertible matrix X such that A = X−1BX. Indeed, repeatedly applying the above identities yields det(A) = det(X)−1 det(B) det(X) = det(B) det(X)−1 det(X) = det(B).

The determinant is therefore also called a similarity invariant. The determinant of a linear transformation

T : V → V for some ﬁnite-dimensional vector space V is deﬁned to be the determinant of the matrix describing it, with respect to an arbitrary choice of basis in V. By the similarity invariance, this determinant is independent of the choice of the basis for V and therefore only depends on the endomorphism T.

4.4.2 Exterior algebra

The determinant of a linear transformation A : V → V of an n-dimensional vector space V can be formulated in a coordinate-free manner by considering the nth exterior power ΛnV of V. A induces a linear map

ΛnA :ΛnV → ΛnV v1 ∧ v2 ∧ · · · ∧ vn 7→ Av1 ∧ Av2 ∧ · · · ∧ Avn. As ΛnV is one-dimensional, the map ΛnA is given by multiplying with some scalar. This scalar coincides with the determinant of A, that is to say

n (Λ A)(v1 ∧ · · · ∧ vn) = det(A) · v1 ∧ · · · ∧ vn.

This deﬁnition agrees with the more concrete coordinate-dependent deﬁnition. This follows from the characterization of the determinant given above. For example, switching two columns changes the sign of the determinant; likewise, permuting the vectors in the exterior product v1 ∧ v2 ∧ v3 ∧ ... ∧ vn to v2 ∧ v1 ∧ v3 ∧ ... ∧ vn, say, also changes its sign. For this reason, the highest non-zero exterior power Λn(V) is sometimes also called the determinant of V and similarly for more involved objects such as vector bundles or chain complexes of vector spaces. Minors of a matrix can also be cast in this setting, by considering lower alternating forms ΛkV with k < n. 20 CHAPTER 4. DETERMINANT

Transformation on alternating multilinear n-forms

The vector space W of all alternating multilinear n-forms on an n-dimensional vector space V has dimension one. To each linear transformation T on V we associate a linear transformation T′ on W, where for each w in W we deﬁne (T′w)(x1, ..., xn) = w(Tx1, ..., Txn). As a linear transformation on a one-dimensional space, T′ is equivalent to a scalar multiple. We call this scalar the determinant of T.

4.4.3 Square matrices over commutative rings and abstract properties

The determinant can also be characterized as the unique function

D : Mn(K) → K

from the set of all n × n matrices with entries in a field K to this field satisfying the following three properties: first, D is an n-linear function: considering all but one column of A fixed, the determinant is linear in the remaining column, that is

D(v1, . . . , vi−1, avi+bw, vi+1, . . . , vn) = aD(v1, . . . , vi−1, vi, vi+1, . . . , vn)+bD(v1, . . . , vi−1, w, vi+1, . . . , vn)

for any column vectors v1, ..., vn, and w and any scalars (elements of K) a and b. Second, D is an alternating function: for any matrix A with two identical columns D(A) = 0. Finally, D(In) = 1. Here In is the identity matrix. This fact also implies that every other n-linear alternating function F:Mn(K) → K satisﬁes

F (M) = F (I)D(M).

This deﬁnition can also be extended where K is a commutative ring R, in which case a matrix is invertible if and only if its determinant is a invertible element in R. For example, a matrix A with entries in Z, the integers, is invertible (in the sense that there exists an inverse matrix with integer entries) if the determinant is +1 or −1. Such a matrix is called unimodular. The determinant deﬁnes a mapping

× GLn(R) → R , between the group of invertible n × n matrices with entries in R and the multiplicative group of units in R. Since it respects the multiplication in both groups, this map is a group homomorphism. Secondly, given a ring homomorphism f: R → S, there is a map GLn(R) → GLn(S) given by replacing all entries in R by their images under f. The determinant respects these maps, i.e., given a matrix A = (ai,j) with entries in R, the identity

f(det((ai,j ))) = det((f(ai,j)))

holds. For example, the determinant of the complex conjugate of a complex matrix (which is also the determinant of its conjugate transpose) is the complex conjugate of its determinant, and for integer matrices: the reduction modulo m of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo m (the latter determinant being computed using modular arithmetic). In the more high-brow parlance of category theory, the determinant is a natural transformation between the two functors GLn and (⋅)×.[11] Adding yet another layer of abstraction, this is captured by saying that the determinant is a morphism of algebraic groups, from the general linear group to the multiplicative group,

det : GLn → Gm. 4.5. GENERALIZATIONS AND RELATED NOTIONS 21

4.5 Generalizations and related notions

4.5.1 Inﬁnite matrices

For matrices with an infinite number of rows and columns, the above definitions of the determinant do not carry over directly. For example, in the Leibniz formula, an infinite sum (all of whose terms are infinite products) would have to be calculated. Functional analysis provides different extensions of the determinant for such infinite-dimensional situations, which however only work for particular kinds of operators. The Fredholm determinant defines the determinant for operators known as trace class operators by an appropriate generalization of the formula det(I + A) = exp(tr(log(I + A))). Another infinite-dimensional notion of determinant is the functional determinant.

4.5.2 Related notions for non-commutative rings

For square matrices with entries in a non-commutative ring, there are various difficulties in defining determinants analogously to that for commutative rings. A meaning can be given to the Leibniz formula provided that the order for the product is specified, and similarly for other ways to define the determinant, but non-commutativity then leads to the loss of many fundamental properties of the determinant, for instance the multiplicative property or the fact that the determinant is unchanged under transposition of the matrix. Over non-commutative rings, there is no reasonable notion of a multilinear form (existence of a nonzero bilinear form with a regular element of R as value on some pair of arguments implies that R is commutative). Nevertheless, various notions of non-commutative determinant have been formulated, which preserve some of the properties of determinants, notably quasideterminants and the Dieudonné determinant. It may be noted that if one considers certain specific classes of matrices with non-commutative elements, then there are examples where one can define the determinant and prove linear algebra theorems that are very similar to their commutative analogs. Examples include quantum groups and q-determinant, Capelli matrix and Capelli determinant, super-matrices and Berezinian; Manin matrices is the class of matrices which is most close to matrices with commutative elements.

4.5.3 Further variants

[12] Determinants of matrices in superrings (that is, Z2-graded rings) are known as Berezinians or superdeterminants. The permanent of a matrix is deﬁned as the determinant, except that the factors sgn(σ) occurring in Leibniz’s rule are omitted. The immanant generalizes both by introducing a character of the symmetric group Sn in Leibniz’s rule.

4.6 Calculation

Determinants are mainly used as a theoretical tool. They are rarely calculated explicitly in numerical linear algebra, where for applications like checking invertibility and finding eigenvalues the determinant has largely been supplanted by other techniques.[13] Nonetheless, explicitly calculating determinants is required in some situations, and different methods are available to do so. Naive methods of implementing an algorithm to compute the determinant include using the Leibniz formula or Laplace’s formula. Both these approaches are extremely inefficient for large matrices, though, since the number of required operations grows very quickly: it is of order n!(n factorial) for an n × n matrix M. For example, Leibniz’s formula requires calculating n! products. Therefore, more involved techniques have been developed for calculating determinants.

4.6.1 Decomposition methods

Given a matrix A, some methods compute its determinant by writing A as a product of matrices whose determinants can be more easily computed. Such techniques are referred to as decomposition methods. Examples include the 22 CHAPTER 4. DETERMINANT

LU decomposition, the QR decomposition or the Cholesky decomposition (for positive deﬁnite matrices). These methods are of order O(n3), which is a signiﬁcant improvement over O(n!) The LU decomposition expresses A in terms of a lower triangular matrix L, an upper triangular matrix U and a permutation matrix P:

A = P LU.

The determinants of L and U can be quickly calculated, since they are the products of the respective diagonal entries. The determinant of P is just the sign ε of the corresponding permutation (which is +1 for an even number of permutations and is −1 for an uneven number of permutations). The determinant of A is then

det(A) = ε det(L) · det(U),

Moreover, the decomposition can be chosen such that L is a unitriangular matrix and therefore has determinant 1, in which case the formula further simpliﬁes to

det(A) = ε det(U).

4.6.2 Further methods

If the determinant of A and the inverse of A have already been computed, the matrix determinant lemma allows to quickly calculate the determinant of A + uvT, where u and v are column vectors. Since the definition of the determinant does not need divisions, a question arises: do fast algorithms exist that do not need divisions? This is especially interesting for matrices over rings. Indeed, algorithms with run-time proportional to n4 exist. An algorithm of Mahajan and Vinay, and Berkowitz[14] is based on closed ordered walks (short clow). It computes more products than the determinant definition requires, but some of these products cancel and the sum of these products can be computed more efficiently. The final algorithm looks very much like an iterated product of triangular matrices. If two matrices of order n can be multiplied in time M(n), where M(n) ≥ na for some a > 2, then the determinant can be computed in time O(M(n)).[15] This means, for example, that an O(n2.376) algorithm exists based on the Coppersmith–Winograd algorithm. Algorithms can also be assessed according to their bit complexity, i.e., how many bits of accuracy are needed to store intermediate values occurring in the computation. For example, the Gaussian elimination (or LU decomposition) methods is of order O(n3), but the bit length of intermediate values can become exponentially long.[16] The Bareiss Algorithm, on the other hand, is an exact-division method based on Sylvester’s identity is also of order n3, but the bit complexity is roughly the bit size of the original entries in the matrix times n.[17]

4.7 History

Historically, determinants were used long before matrices: originally, a determinant was defined as a property of a system of linear equations. The determinant “determines” whether the system has a unique solution (which occurs precisely if the determinant is non-zero). In this sense, determinants were first used in the Chinese mathematics textbook The Nine Chapters on the Mathematical Art (, Chinese scholars, around the 3rd century BCE). In Europe, 2 × 2 determinants were considered by Cardano at the end of the 16th century and larger ones by Leibniz.[18][19][20][21] In Japan, Seki Takakazu ( ) is credited with the discovery of the resultant and the determinant (at first in 1683, the complete version no later than 1710). In Europe, Cramer (1750) added to the theory, treating the subject in relation to sets of equations. The recurrence law was first announced by Bézout (1764). It was Vandermonde (1771) who first recognized determinants as independent functions.[18] Laplace (1772) [22][23] gave the general method of expanding a determinant in terms of its complementary minors: Vandermonde had already given a special case. Immediately following, Lagrange (1773) treated determinants of the second and third order and applied it to questions of elimination theory; he proved many special cases of general identities. 4.8. APPLICATIONS 23

Gauss (1801) made the next advance. Like Lagrange, he made much use of determinants in the theory of numbers. He introduced the word determinant (Laplace had used resultant), though not in the present signification, but rather as applied to the discriminant of a quantic. Gauss also arrived at the notion of reciprocal (inverse) determinants, and came very near the multiplication theorem. The next contributor of importance is Binet (1811, 1812), who formally stated the theorem relating to the product of two matrices of m columns and n rows, which for the special case of m = n reduces to the multiplication theorem. On the same day (November 30, 1812) that Binet presented his paper to the Academy, Cauchy also presented one on the subject. (See Cauchy–Binet formula.) In this he used the word determinant in its present sense,[24][25] summarized and simplified what was then known on the subject, improved the notation, and gave the multiplication theorem with a proof more satisfactory than Binet’s.[18][26] With him begins the theory in its generality. The next important figure was Jacobi[19] (from 1827). He early used the functional determinant which Sylvester later called the Jacobian, and in his memoirs in Crelle for 1841 he specially treats this subject, as well as the class of alternating functions which Sylvester has called alternants. About the time of Jacobi’s last memoirs, Sylvester (1839) and Cayley began their work.[27][28] The study of special forms of determinants has been the natural result of the completion of the general theory. Axisymmetric determinants have been studied by Lebesgue, Hesse, and Sylvester; persymmetric determinants by Sylvester and Hankel; circulants by Catalan, Spottiswoode, Glaisher, and Scott; skew determinants and Pfaffians, in connection with the theory of orthogonal transformation, by Cayley; continuants by Sylvester; Wronskians (so called by Muir) by Christoffel and Frobenius; compound determinants by Sylvester, Reiss, and Picquet; Jacobians and Hessians by Sylvester; and symmetric gauche determinants by Trudi. Of the textbooks on the subject Spottiswoode’s was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises.

4.8 Applications

4.8.1 Linear independence

As mentioned above, the determinant of a matrix (with real or complex entries, say) is zero if and only if the column vectors (or the row vectors) of the matrix are linearly dependent. Thus, determinants can be used to characterize 3 linearly dependent vectors. For example, given two linearly independent vectors v1, v2 in R , a third vector v3 lies in the plane spanned by the former two vectors exactly if the determinant of the 3 × 3 matrix consisting of the three vectors is zero. The same idea is also used in the theory of differential equations: given n functions f1(x), ..., fn(x) (supposed to be n − 1 times differentiable), the Wronskian is defined to be

··· f1(x) f2(x) fn(x) ′ ′ ··· ′ f1(x) f2(x) fn(x)

W (f1, . . . , fn)(x) = ...... (n−1) (n−1) ··· (n−1) f1 (x) f2 (x) fn (x)

It is non-zero (for some x) in a speciﬁed interval if and only if the given functions and all their derivatives up to order n−1 are linearly independent. If it can be shown that the Wronskian is zero everywhere on an interval then, in the case of analytic functions, this implies the given functions are linearly dependent. See the Wronskian and linear independence.

4.8.2 Orientation of a basis

Main article: Orientation (vector space)

The determinant can be thought of as assigning a number to every sequence of n vectors in Rn, by using the square matrix whose columns are the given vectors. For instance, an orthogonal matrix with entries in Rn represents an orthonormal basis in Euclidean space. The determinant of such a matrix determines whether the orientation of the basis is consistent with or opposite to the orientation of the standard basis. If the determinant is +1, the basis has the same orientation. If it is −1, the basis has the opposite orientation. 24 CHAPTER 4. DETERMINANT

More generally, if the determinant of A is positive, A represents an orientation-preserving linear transformation (if A is an orthogonal 2 × 2 or 3 × 3 matrix, this is a rotation), while if it is negative, A switches the orientation of the basis.

4.8.3 Volume and Jacobian determinant

As pointed out above, the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors. As a consequence, if f: Rn → Rn is the linear map represented by the matrix A, and S is any measurable subset of Rn, then the volume of f(S) is given by |det(A)| times the volume of S. More generally, if the linear map f: Rn → Rm is represented by the m × n matrix A, then the n-dimensional volume of f(S) is given by:

√ volume(f(S)) = det(ATA) × volume(S).

By calculating the volume of the tetrahedron bounded by four points, they can be used to identify skew lines. The volume of any tetrahedron, given its vertices a, b, c, and d, is (1/6)·|det(a − b, b − c, c − d)|, or any other combination of pairs of vertices that would form a spanning tree over the vertices. For a general diﬀerentiable function, much of the above carries over by considering the Jacobian matrix of f. For

f : Rn → Rn,

the Jacobian is the n × n matrix whose entries are given by

( ) ∂f D(f) = i . ∂xj 1≤i,j≤n Its determinant, the Jacobian determinant appears in the higher-dimensional version of integration by substitution: for suitable functions f and an open subset U of R'n (the domain of f), the integral over f(U) of some other function φ: Rn → Rm is given by

∫ ∫ ϕ(v) dv = ϕ(f(u)) |det(D f)(u)| du. f(U) U The Jacobian also occurs in the inverse function theorem.

4.8.4 Vandermonde determinant (alternant)

Main article: Vandermonde matrix

Third order

1 1 1

x1 x2 x3 = (x3 − x2)(x3 − x1)(x2 − x1) . 2 2 2 x1 x2 x3 In general, the nth-order Vandermonde determinant is [29]

1 1 1 ··· 1 ··· x1 x2 x3 xn ∏ 2 2 2 ··· 2 x1 x2 x3 xn − = (xj xi) , ...... ≤ ≤ . . . . . 1 i

4.8.5 Circulants

Main article: Circulant matrix

Second order

x1 x2 = (x1 + x2)(x1 − x2) . x2 x1 Third order

x1 x2 x3 ( )( ) 2 2 x3 x1 x2 = (x1 + x2 + x3) x1 + ωx2 + ω x3 x1 + ω x2 + ωx3 ,

x2 x3 x1 where ω and ω2 are the complex cube roots of 1. In general, the nth-order circulant determinant is[29]

x1 x2 x3 ··· xn

x x x ··· x − n 1 2 n 1 ∏n ( ) ··· − xn−1 xn x1 xn−2 = x + x ω + x ω2 + ... + x ωn 1 , . . . . 1 2 j 3 j n j ...... j=1 . . . . .

x2 x3 x4 ··· x1 where ωj is an nth root of 1.

4.9 See also

• Dieudonné determinant • Functional determinant • Immanant • Matrix determinant lemma • Permanent • Pfaﬃan • Slater determinant

4.10 Notes

[1] Serge Lang, Linear Algebra, 2nd Edition, Addison-Wesley, 1971, pp 173, 191. [2] WildLinAlg episode 4, Norman J Wildberger, Univ. of New South Wales, 2010, lecture via youtube [3] In a non-commutative setting left-linearity (compatibility with left-multiplication by scalars) should be distinguished from right-linearity. Assuming linearity in the columns is taken to be left-linearity, one would have, for non-commuting scalars a, b:

1 0 1 0 a 0 a 0 1 0 ab = ab = a = = b = ba = ba, 0 1 0 b 0 b 0 1 0 1 a contradiction. There is no useful notion of multi-linear functions over a non-commutative ring. 26 CHAPTER 4. DETERMINANT

[4] §0.8.2 of R. A. Horn & C. R. Johnson: Matrix Analysis 2nd ed. (2013) Cambridge University Press. ISBN 9780521548236.

[5] Proofs can be found in http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/proof003.html

[6] A proof can be found in the Appendix B of Kondratyuk, L. A.; Krivoruchenko, M. I. (1992). “Superconducting quark matter in SU(2) color group”. Zeitschrift für Physik A 344: 99–115. doi:10.1007/BF01291027.

[7] Ken Habgood, Itamar Arel, A condensation-based application of Cramerʼs rule for solving large-scale linear systems, Journal of Discrete Algorithms, 10 (2012), pp. 98–109. Available online 1 July 2011, ISSN 1570–8667, 10.1016/j.jda.2011.06.007.

[8] These identities were taken from http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/proof003.html

[9] Proofs are given in J.R. Silvester, Determinants of Block Matrices, Math. Gazette, 84 (2000), pp. 460–467, available at http://www.jstor.org/stable/3620776 or freely at http://www.ee.iisc.ernet.in/new/people/faculty/prasantg/downloads/ blocks.pdf

[10] §0.8.10 of R. A. Horn & C. R. Johnson: Matrix Analysis 2nd ed. (2013) Cambridge University Press. ISBN 9780521548236.

[11] Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics 5 ((2nd ed.) ed.), Springer-Verlag, ISBN 0-387-98403-8

[12] Varadarajan, V. S (2004), Supersymmetry for mathematicians: An introduction, ISBN 978-0-8218-3574-6.

[13] L. N. Trefethen and D. Bau, Numerical Linear Algebra (SIAM, 1997). e.g. in Lecture 1: "... we mention that the determinant, though a convenient notion theoretically, rarely ﬁnds a useful role in numerical algorithms.”

[14] http://page.inf.fu-berlin.de/~{}rote/Papers/pdf/Division-free+algorithms.pdf

[15] Bunch, J. R.; Hopcroft, J. E. (1974). “Triangular Factorization and Inversion by Fast Matrix Multiplication”. Mathematics of Computation 28 (125): 231–236. doi:10.1090/S0025-5718-1974-0331751-8.

[16] Fang, Xin Gui; Havas, George (1997). “On the worst-case complexity of integer Gaussian elimination” (PDF). Proceedings of the 1997 international symposium on Symbolic and algebraic computation. ISSAC '97. Kihei, Maui, Hawaii, United States: ACM. pp. 28–31. doi:10.1145/258726.258740. ISBN 0-89791-875-4.

[17] Bareiss, Erwin (1968), “Sylvester’s Identity and Multistep Integer-Preserving Gaussian Elimination” (PDF), Mathematics of computation 22 (102): 565–578

[18] Campbell, H: “Linear Algebra With Applications”, pages 111–112. Appleton Century Crofts, 1971

[19] Eves, H: “An Introduction to the History of Mathematics”, pages 405, 493–494, Saunders College Publishing, 1990.

[20] A Brief History of Linear Algebra and Matrix Theory : http://darkwing.uoregon.edu/~{}vitulli/441.sp04/LinAlgHistory. html

[21] Cajori, F. A History of Mathematics p. 80

[22] Expansion of determinants in terms of minors: Laplace, Pierre-Simon (de) “Researches sur le calcul intégral et sur le systéme du monde,” Histoire de l'Académie Royale des Sciences (Paris), seconde partie, pages 267–376 (1772).

[23] Muir, Sir Thomas, The Theory of Determinants in the historical Order of Development [London, England: Macmillan and Co., Ltd., 1906]. JFM 37.0181.02

[24] The ﬁrst use of the word “determinant” in the modern sense appeared in: Cauchy, Augustin-Louis “Memoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et des signes contraires par suite des transpositions operées entre les variables qu'elles renferment,” which was ﬁrst read at the Institute de France in Paris on November 30, 1812, and which was subsequently published in the Journal de l'Ecole Polytechnique, Cahier 17, Tome 10, pages 29–112 (1815).

[25] Origins of mathematical terms: http://jeff560.tripod.com/d.html

[26] History of matrices and determinants: http://www-history.mcs.st-and.ac.uk/history/HistTopics/Matrices_and_determinants. html

[27] The ﬁrst use of vertical lines to denote a determinant appeared in: Cayley, Arthur “On a theorem in the geometry of position,” Cambridge Mathematical Journal, vol. 2, pages 267–271 (1841).

[28] History of matrix notation: http://jeff560.tripod.com/matrices.html

[29] Gradshteyn, I. S., I. M. Ryzhik: “Table of Integrals, Series, and Products”, 14.31, Elsevier, 2007. 4.11. REFERENCES 27

4.11 References

See also: Linear algebra § Further reading

• Axler, Sheldon Jay (1997), Linear Algebra Done Right (2nd ed.), Springer-Verlag, ISBN 0-387-98259-0

• de Boor, Carl (1990), “An empty exercise” (PDF), ACM SIGNUM Newsletter 25 (2): 3–7, doi:10.1145/122272.122273. • Lay, David C. (August 22, 2005), Linear Algebra and Its Applications (3rd ed.), Addison Wesley, ISBN 978- 0-321-28713-7 • Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), ISBN 978-0-89871-454-8 • Muir, Thomas (1960) [1933], A treatise on the theory of determinants, Revised and enlarged by William H. Metzler, New York, NY: Dover • Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Brooks/Cole, ISBN 0-534-99845-3

• Horn, R. A.; Johnson, C. R. (2013), Matrix Analysis (2nd ed.), Cambridge University Press, ISBN 9780521548236 • Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International

• Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall

4.12 External links

• Hazewinkel, Michiel, ed. (2001), “Determinant”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608- 010-4

• Weisstein, Eric W., “Determinant”, MathWorld. • O'Connor, John J.; Robertson, Edmund F., “Matrices and determinants”, MacTutor History of Mathematics archive, University of St Andrews. • WebApp to calculate determinants and descriptively solve systems of linear equations

• Determinant Interactive Program and Tutorial • Online Matrix Calculator

• Linear algebra: determinants. Compute determinants of matrices up to order 6 using Laplace expansion you choose.

• Matrices and Linear Algebra on the Earliest Uses Pages • Determinants explained in an easy fashion in the 4th chapter as a part of a Linear Algebra course.

• Instructional Video on taking the determinant of an nxn matrix (Khan Academy)

• Online matrix calculator (determinant, track, inverse, adjoint, transpose) Compute determinant of matrix up to order 8

• Derivation of Determinant of a Matrix Chapter 5

Greatest common divisor

In mathematics, the greatest common divisor (gcd) of two or more integers, when at least one of them is not zero, is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4.[1][2] The GCD is also known as the greatest common factor (gcf),[3] highest common factor (hcf),[4] greatest common measure (gcm),[5] or highest common divisor.[6] This notion can be extended to polynomials (see Polynomial greatest common divisor) and other commutative rings (see below).

5.1 Overview

5.1.1 Notation

In this article we will denote the greatest common divisor of two integers a and b as gcd(a,b). Some textbooks use (a,b).[1][2][6][7] The J (programming language) uses a +. b

5.1.2 Example

The number 54 can be expressed as a product of two integers in several diﬀerent ways:

54 × 1 = 27 × 2 = 18 × 3 = 9 × 6.

Thus the divisors of 54 are:

1, 2, 3, 6, 9, 18, 27, 54.

Similarly the divisors of 24 are:

1, 2, 3, 4, 6, 8, 12, 24.

The numbers that these two lists share in common are the common divisors of 54 and 24:

1, 2, 3, 6.

The greatest of these is 6. That is the greatest common divisor of 54 and 24. One writes:

28 5.2. CALCULATION 29

gcd(54, 24) = 6.

5.1.3 Reducing fractions

The greatest common divisor is useful for reducing fractions to be in lowest terms. For example, gcd(42, 56) = 14, therefore,

42 3 · 14 3 = = . 56 4 · 14 4

5.1.4 Coprime numbers

Two numbers are called relatively prime, or coprime, if their greatest common divisor equals 1. For example, 9 and 28 are relatively prime.

5.1.5 A geometric view

For example, a 24-by-60 rectangular area can be divided into a grid of: 1-by-1 squares, 2-by-2 squares, 3-by-3 squares, 4-by-4 squares, 6-by-6 squares or 12-by-12 squares. Therefore, 12 is the greatest common divisor of 24 and 60. A 24-by-60 rectangular area can be divided into a grid of 12-by-12 squares, with two squares along one edge (24/12 = 2) and ﬁve squares along the other (60/12 = 5).

5.2 Calculation

5.2.1 Using prime factorizations

Greatest common divisors can in principle be computed by determining the prime factorizations of the two numbers and comparing factors, as in the following example: to compute gcd(18, 84), we find the prime factorizations 18 = 2 · 32 and 84 = 22 · 3 · 7 and notice that the “overlap” of the two expressions is 2 · 3; so gcd(18, 84) = 6. In practice, this method is only feasible for small numbers; computing prime factorizations in general takes far too long. Here is another concrete example, illustrated by a Venn diagram. Suppose it is desired to find the greatest common divisor of 48 and 180. First, find the prime factorizations of the two numbers:

48 = 2 × 2 × 2 × 2 × 3, 180 = 2 × 2 × 3 × 3 × 5.

What they share in common is two “2"s and a “3":

2 2 3 2 2 5 3 30 CHAPTER 5. GREATEST COMMON DIVISOR

Least common multiple = 2 × 2 × ( 2 × 2 × 3 ) × 3 × 5 = 720 Greatest common divisor = 2 × 2 × 3 = 12.

5.2.2 Using Euclid’s algorithm

A much more eﬃcient method is the Euclidean algorithm, which uses a division algorithm such as long division in combination with the observation that the gcd of two numbers also divides their diﬀerence. To compute gcd(48,18), divide 48 by 18 to get a quotient of 2 and a remainder of 12. Then divide 18 by 12 to get a quotient of 1 and a remainder of 6. Then divide 12 by 6 to get a remainder of 0, which means that 6 is the gcd. Note that we ignored the quotient in each step except to notice when the remainder reached 0, signalling that we had arrived at the answer. Formally the algorithm can be described as:

gcd(a, 0) = a

gcd(a, b) = gcd(b, a mod b) where

⌊ ⌋ a a mod b = a − b b If the arguments are both greater than zero then the algorithm can be written in more elementary terms as follows:

gcd(a, a) = a gcd(a, b) = gcd(a − b, b) , if a > b gcd(a, b) = gcd(a, b − a) , if b > a

Complexity of Euclidean method

The existence of the Euclidean algorithm places (the decision problem version of) the greatest common divisor problem in P, the class of problems solvable in polynomial time. The GCD problem is not known to be in NC, and so there is no known way to parallelize its computation across many processors; nor is it known to be P-complete, which would imply that it is unlikely to be possible to parallelize GCD computation. In this sense the GCD problem is analogous to e.g. the integer factorization problem, which has no known polynomial-time algorithm, but is not known to be NP-complete. Shallcross et al. showed that a related problem (EUGCD, determining the remainder sequence arising during the Euclidean algorithm) is NC-equivalent to the problem of integer linear programming with two variables; if either problem is in NC or is P-complete, the other is as well.[8] Since NC contains NL, it is also unknown whether a space-eﬃcient algorithm for computing the GCD exists, even for nondeterministic Turing machines. Although the problem is not known to be in NC, parallel algorithms asymptotically faster than the Euclidean algorithm exist; the best known deterministic algorithm is by Chor and Goldreich, which (in the CRCW-PRAM model) can 1+ε [9] solve the problem[ in( O(√n/log n))] time with n processors. Randomized algorithms can solve the problem in O((log n)2) time on exp O n log n processors (note this is superpolynomial).[10]

5.2.3 Binary method

An alternative method of computing the gcd is the binary gcd method which uses only subtraction and division by 2. In outline the method is as follows: Let a and b be the two non negative integers. Also set the integer d to 0. There are ﬁve possibilities:

• a = b.

As gcd(a, a) = a, the desired gcd is a×2d (as a and b are changed in the other cases, and d records the number of times that a and b have been both divided by 2 in the next step, the gcd of the initial pair is the product of a by 2d). 5.2. CALCULATION 31

• Both a and b are even.

In this case 2 is a common divisor. Divide both a and b by 2, increment d by 1 to record the number of times 2 is a common divisor and continue.

• a is even and b is odd.

In this case 2 is not a common divisor. Divide a by 2 and continue.

• a is odd and b is even.

As in the previous case 2 is not a common divisor. Divide b by 2 and continue.

• Both a and b are odd.

As gcd(a,b) = gcd(b,a) and we have already considered the case a = b, we may assume that a > b. The number c = a − b is smaller than a yet still positive. Any number that divides a and b must also divide c so every common divisor of a and b is also a common divisor of b and c Similarly, a = b + c and every common divisor of b and c is also a common divisor of a and b. So the two pairs (a, b) and (b, c) have the same common divisors, and thus gcd(a,b) = gcd(b,c). Moreover, as a and b are both odd, c is even, and one may replace c by c/2 without changing the gcd. Thus the process can be continued with the pair (a, b) replaced by the smaller numbers (c/2, b). Each of the above steps reduces at least one of a and b towards 0 and so can only be repeated a ﬁnite number of times. Thus one must eventually reach the case a = b, which is the only stopping case. Then, as quoted above, the gcd is a×2d. This algorithm may easily programmed as follows: Input: a, b positive integers Output: g and d such that g is odd and gcd(a, b) = g×2d d := 0 while a and b are both even do a := a/2 b := b/2 d := d + 1 while a ≠ b do if a is even then a := a/2 else if b is even then b := b/2 else if a > b then a := (a – b)/2 else b := (b – a)/2 g := a output g, d Example: (a, b, d) = (48, 18, 0) → (24, 9, 1) → (12, 9, 1) → (6, 9, 1) → (3, 9, 1) → (3, 6, 1) → (3, 3, 1) ; the original gcd is thus 2d = 21 times a= b= 3, that is 6. The Binary GCD algorithm is particularly easy to implement on binary computers. The test for whether a number is divisible by two can be performed by testing the lowest bit in the number. Division by two can be achieved by shifting the input number by one bit. Each step of the algorithm makes at least one such shift. Subtracting two numbers smaller than a and b costs O(log a + log b) bit operations. Each step makes at most one such subtraction. The total number of steps is at most the sum of the numbers of bits of a and b, hence the computational complexity is

O((log a + log b)2) For further details see Binary GCD algorithm.

5.2.4 Other methods

If a and b are both nonzero, the greatest common divisor of a and b can be computed by using least common multiple (lcm) of a and b:

a · b gcd(a, b) = lcm(a, b) but more commonly the lcm is computed from the gcd. Using Thomae’s function f,

( ) b gcd(a, b) = af , a 32 CHAPTER 5. GREATEST COMMON DIVISOR

which generalizes to a and b rational numbers or commensurable real numbers. Keith Slavin has shown that for odd a ≥ 1:

a∏−1 −2iπkb/a gcd(a, b) = log2 (1 + e ) k=0 which is a function that can be evaluated for complex b.[11] Wolfgang Schramm has shown that

∑a ∑ c (k) gcd(a, b) = exp(2πikb/a) · d d k=1 d|a

is an entire function in the variable b for all positive integers a where cd(k) is Ramanujan’s sum.[12] Donald Knuth proved the following reduction:

gcd(2a − 1, 2b − 1) = 2gcd(a,b) − 1

for non-negative integers a and b, where a and b are not both zero.[13] More generally

gcd(na − 1, nb − 1) = ngcd(a,b) − 1

which can be proven by considering the Euclidean algorithm in base n. Another useful identity relates gcd(a, b) to the Euler’s totient function:

∑ gcd(a, b) = φ(k). k|a and k|b

5.3 Properties

• Every common divisor of a and b is a divisor of gcd(a, b). • gcd(a, b), where a and b are not both zero, may be deﬁned alternatively and equivalently as the smallest positive integer d which can be written in the form d = a·p + b·q, where p and q are integers. This expression is called Bézout’s identity. Numbers p and q like this can be computed with the extended Euclidean algorithm. • gcd(a, 0) = |a|, for a ≠ 0, since any number is a divisor of 0, and the greatest divisor of a is |a|.[2][6] This is usually used as the base case in the Euclidean algorithm. • If a divides the product b·c, and gcd(a, b) = d, then a/d divides c. • If m is a non-negative integer, then gcd(m·a, m·b) = m·gcd(a, b). • If m is any integer, then gcd(a + m·b, b) = gcd(a, b). • If m is a nonzero common divisor of a and b, then gcd(a/m, b/m) = gcd(a, b)/m.

• The gcd is a multiplicative function in the following sense: if a1 and a2 are relatively prime, then gcd(a1·a2, b) = gcd(a1, b)·gcd(a2, b). In particular, recalling that gcd is a positive integer valued function (i.e., gets natural values only) we obtain that gcd(a, b·c) = 1 if and only if gcd(a, b) = 1 and gcd(a, c) = 1. • The gcd is a commutative function: gcd(a, b) = gcd(b, a). • The gcd is an associative function: gcd(a, gcd(b, c)) = gcd(gcd(a, b), c). • The gcd of three numbers can be computed as gcd(a, b, c) = gcd(gcd(a, b), c), or in some diﬀerent way by applying commutativity and associativity. This can be extended to any number of numbers. 5.4. PROBABILITIES AND EXPECTED VALUE 33

• gcd(a, b) is closely related to the least common multiple lcm(a, b): we have

gcd(a, b)·lcm(a, b) = a·b. This formula is often used to compute least common multiples: one ﬁrst computes the gcd with Euclid’s algorithm and then divides the product of the given numbers by their gcd.

• The following versions of distributivity hold true:

gcd(a, lcm(b, c)) = lcm(gcd(a, b), gcd(a, c)) lcm(a, gcd(b, c)) = gcd(lcm(a, b), lcm(a, c)).

• It is sometimes useful to deﬁne gcd(0, 0) = 0 and lcm(0, 0) = 0 because then the natural numbers become a complete distributive lattice with gcd as meet and lcm as join operation.[14] This extension of the deﬁnition is also compatible with the generalization for commutative rings given below.

• In a Cartesian coordinate system, gcd(a, b) can be interpreted as the number of segments between points with integral coordinates on the straight line segment joining the points (0, 0) and (a, b).

5.4 Probabilities and expected value

In 1972, James E. Nymann showed that k integers, chosen independently and uniformly from {1,...,n}, are coprime with probability 1/ζ(k) as n goes to inﬁnity, where ζ refers to the Riemann zeta function.[15] (See coprime for a derivation.) This result was extended in 1987 to show that the probability that k random integers have greatest common divisor d is d−k/ζ(k).[16] Using this information, the expected value of the greatest common divisor function can be seen (informally) to not exist when k = 2. In this case the probability that the gcd equals d is d−2/ζ(2), and since ζ(2) = π2/6 we have

∞ ∞ ∑ 6 6 ∑ 1 E(2) = d = . π2d2 π2 d d=1 d=1 This last summation is the harmonic series, which diverges. However, when k ≥ 3, the expected value is well-deﬁned, and by the above argument, it is

∞ ∑ ζ(k − 1) E(k) = d1−kζ(k)−1 = . ζ(k) d=1 For k = 3, this is approximately equal to 1.3684. For k = 4, it is approximately 1.1106.

5.5 The gcd in commutative rings

See also: divisor (ring theory)

The notion of greatest common divisor can more generally be defined for elements of an arbitrary commutative ring, although in general there need not exist one for every pair of elements. If R is a commutative ring, and a and b are in R, then an element d of R is called a common divisor of a and b if it divides both a and b (that is, if there are elements x and y in R such that d·x = a and d·y = b). If d is a common divisor of a and b, and every common divisor of a and b divides d, then d is called a greatest common divisor of a and b. Note that with this definition, two elements a and b may very well have several greatest common divisors, or none at all. If R is an integral domain then any two gcd’s of a and b must be associate elements, since by definition either one must divide the other; indeed if a gcd exists, any one of its associates is a gcd as well. Existence of a gcd is not 34 CHAPTER 5. GREATEST COMMON DIVISOR assured in arbitrary integral domains. However if R is a unique factorization domain, then any two elements have a gcd, and more generally this is true in gcd domains. If R is a Euclidean domain in which euclidean division is given algorithmically (as is the case for instance when R = F[X] where F is a field, or when R is the ring of Gaussian integers), then greatest common divisors can be computed using a form of the Euclidean algorithm based on the division procedure. The following is an example of an integral domain with two elements that do not have a gcd:

[√ ] ( √ )( √ ) ( √ ) R = Z −3 , a = 4 = 2 · 2 = 1 + −3 1 − −3 , b = 1 + −3 · 2.

The elements 2 and 1 + √(−3) are two “maximal common divisors” (i.e. any common divisor which is a multiple of 2 is associated to 2, the same holds for 1 + √(−3)), but they are not associated, so there is no greatest common divisor of a and b. Corresponding to the Bézout property we may, in any commutative ring, consider the collection of elements of the form pa + qb, where p and q range over the ring. This is the ideal generated by a and b, and is denoted simply (a, b). In a ring all of whose ideals are principal (a principal ideal domain or PID), this ideal will be identical with the set of multiples of some ring element d; then this d is a greatest common divisor of a and b. But the ideal (a, b) can be useful even when there is no greatest common divisor of a and b. (Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat’s Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.)

5.6 See also

• Binary GCD algorithm • Coprime • Euclidean algorithm • Extended Euclidean algorithm • Least common multiple • Lowest common denominator • Maximal common divisor • Polynomial greatest common divisor • Bezout domain

5.7 Notes

[1] Long (1972, p. 33)

[2] Pettofrezzo & Byrkit (1970, p. 34)

[3] Kelley, W. Michael (2004), The Complete Idiot’s Guide to Algebra, Penguin, p. 142, ISBN 9781592571611.

[4] Jones, Allyn (1999), Whole Numbers, Decimals, Percentages and Fractions Year 7, Pascal Press, p. 16, ISBN 9781864413786.

[5] Barlow, Peter; Peacock, George; Lardner, Dionysius; Airy, Sir George Biddell; Hamilton, H. P.; Levy, A.; De Morgan, Augustus; Mosley, Henry (1847), Encyclopaedia of Pure Mathematics, R. Griﬃn and Co., p. 589.

[6] Hardy & Wright (1979, p. 20)

[7] Andrews (1994, p. 16) explains his choice of notation: “Many authors write (a,b) for g.c.d.(a,b). We do not, because we shall often use (a,b) to represent a point in the Euclidean plane.”

[8] Shallcross, D.; Pan, V.; Lin-Kriz, Y. (1993). “The NC equivalence of planar integer linear programming and Euclidean GCD” (PDF). 34th IEEE Symp. Foundations of Computer Science. pp. 557–564. 5.8. REFERENCES 35

[9] Chor, B.; Goldreich, O. (1990). “An improved parallel algorithm for integer GCD”. Algorithmica 5 (1–4): 1–10. doi:10.1007/BF01840374.

[10] Adleman, L. M.; Kompella, K. (1988). “Using smoothness to achieve parallelism”. 20th Annual ACM Symposium on Theory of Computing. New York. pp. 528–538. doi:10.1145/62212.62264. ISBN 0-89791-264-0.

[11] Slavin, Keith R. (2008). “Q-Binomials and the Greatest Common Divisor”. Integers Electronic Journal of Combinatorial Number Theory (University of West Georgia, Charles University in Prague) 8: A5. Retrieved 2008-05-26.

[12] Schramm, Wolfgang (2008). “The Fourier transform of functions of the greatest common divisor”. Integers Electronic Journal of Combinatorial Number Theory (University of West Georgia, Charles University in Prague) 8: A50. Retrieved 2008-11-25.

[13] Knuth, Donald E.; Graham, R. L.; Patashnik, O. (March 1994). Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley. ISBN 0-201-55802-5.

[14] Müller-Hoissen, Folkert; Walther, Hans-Otto (2012), “Dov Tamari (formerly Bernhard Teitler)", in Müller-Hoissen, Folk- ert; Pallo, Jean Marcel; Stasheﬀ, Jim, Associahedra, Tamari Lattices and Related Structures: Tamari Memorial Festschrift, Progress in Mathematics 299, Birkhäuser, pp. 1–40, ISBN 9783034804059. Footnote 27, p. 9: “For example, the natural numbers with gcd (greatest common divisor) as meet and lcm (least common multiple) as join operation determine a (complete distributive) lattice.” Including these deﬁnitions for 0 is necessary for this result: if one instead omits 0 from the set of natural numbers, the resulting lattice is not complete.

[15] Nymann, J. E. (1972). “On the probability that k positive integers are relatively prime”. Journal of Number Theory 4 (5): 469–473. doi:10.1016/0022-314X(72)90038-8.

[16] Chidambaraswamy, J.; Sitarmachandrarao, R. (1987). “On the probability that the values of m polynomials have a given g.c.d.”. Journal of Number Theory 26 (3): 237–245. doi:10.1016/0022-314X(87)90081-3.

5.8 References

• Andrews, George E. (1994) [1971], Number Theory, Dover, ISBN 9780486682525 • Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (Fifth ed.), Oxford: Oxford University Press, ISBN 978-0-19-853171-5

• Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77171950

• Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliﬀs: Prentice Hall, LCCN 71081766

5.9 Further reading

• Donald Knuth. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89684-2. Section 4.5.2: The Greatest Common Divisor, pp. 333–356.

• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Cliﬀord Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 31.2: Greatest common divisor, pp. 856–862. • Saunders MacLane and Garrett Birkhoﬀ. A Survey of Modern Algebra, Fourth Edition. MacMillan Publishing Co., 1977. ISBN 0-02-310070-2. 1–7: “The Euclidean Algorithm.”

5.10 External links

• greatest common divisor at Everything2.com • Greatest Common Measure: The Last 2500 Years, by Alexander Stepanov 36 CHAPTER 5. GREATEST COMMON DIVISOR

A 24-by-60 rectangle is covered with ten 12-by-12 square tiles, where 12 is the GCD of 24 and 60. More generally, an a-by-b rectangle can be covered with square tiles of side length c only if c is a common divisor of a and b. 5.10. EXTERNAL LINKS 37

Animation showing an application of the Euclidean Algorithm to ﬁnd the Great Common Divisor of 62 and 36 which is 2. Chapter 6

Linear equation

Graph sample of linear equations.

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the ﬁrst power of) a single variable. Linear equations can have one or more variables. Linear equations occur abundantly in most subareas of mathematics

38 6.1. ONE VARIABLE 39

and especially in applied mathematics. While they arise quite naturally when modeling many phenomena, they are particularly useful since many non-linear equations may be reduced to linear equations by assuming that quantities of interest vary to only a small extent from some “background” state. Linear equations do not include exponents. This article considers the case of a single equation for which one searches the real solutions. All its content applies for complex solutions and, more generally for linear equations with coeﬃcients and solutions in any ﬁeld.

6.1 One variable

A linear equation in one unknown x may always be rewritten

ax = b.

If a ≠ 0, there is a unique solution

b x = . a If a = 0, then either the equation does not have any solution, if b ≠ 0 (it is inconsistent), or every number is a solution, if b is also zero.

6.2 Two variables

A common form of a linear equation in the two variables x and y is y = mx + b, where m and b designate constants (parameters). The origin of the name “linear” comes from the fact that the set of solutions of such an equation forms a straight line in the plane. In this particular equation, the constant m determines the slope or gradient of that line, and the constant term b determines the point at which the line crosses the y-axis, otherwise known as the y-intercept. Since terms of linear equations cannot contain products of distinct or equal variables, nor any power (other than 1) or other function of a variable, equations involving terms such as xy, x2, y1/3, and sin(x) are nonlinear.

6.2.1 Forms for two-dimensional linear equations

Linear equations can be rewritten using the laws of elementary algebra into several diﬀerent forms. These equations are often referred to as the “equations of the straight line.” In what follows, x, y, t, and θ are variables; other letters represent constants (ﬁxed numbers).

General (or standard) form

In the general (or standard[1]) form the linear equation is written as:

Ax + By = C,

where A and B are not both equal to zero. The equation is usually written so that A ≥ 0, by convention. The graph of the equation is a straight line, and every straight line can be represented by an equation in the above form. If A is nonzero, then the x-intercept, that is, the x-coordinate of the point where the graph crosses the x-axis (where, y is zero), is C/A. If B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis (where x is zero), is C/B, and the slope of the line is −A/B. The general form is sometimes written as: 40 CHAPTER 6. LINEAR EQUATION

ax + by + c = 0,

where a and b are not both equal to zero. The two versions can be converted from one to the other by moving the constant term to the other side of the equal sign.

Slope–intercept form

y = mx + b, where m is the slope of the line and b is the y intercept, which is the y coordinate of the location where the line crosses the y axis. This can be seen by letting x = 0, which immediately gives y = b. It may be helpful to think about this in terms of y = b + mx; where the line passes through the point (0, b) and extends to the left and right at a slope of m. Vertical lines, having undeﬁned slope, cannot be represented by this form.

Point–slope form

y − y1 = m(x − x1), where m is the slope of the line and (x1,y1) is any point on the line. The point-slope form expresses the fact that the diﬀerence in the y coordinate between two points on a line (that is, y − y1) is proportional to the diﬀerence in the x coordinate (that is, x − x1). The proportionality constant is m (the slope of the line).

Two-point form

y2 − y1 y − y1 = (x − x1), x2 − x1

where (x1, y1) and (x2, y2) are two points on the line with x2 ≠ x1. This is equivalent to the point-slope form above, where the slope is explicitly given as (y2 − y1)/(x2 − x1).

Multiplying both sides of this equation by (x2 − x1) yields a form of the line generally referred to as the symmetric form:

(x2 − x1)(y − y1) = (y2 − y1)(x − x1).

Expanding the products and regrouping the terms leads to the general form:

x (y2 − y1) − y (x2 − x1) = x1y2 − x2y1

Using a determinant, one gets a determinant form, easy to remember:

x y 1

x1 y1 1 = 0 .

x2 y2 1

Intercept form x y + = 1, a b where a and b must be nonzero. The graph of the equation has x-intercept a and y-intercept b. The intercept form is in standard form with A/C = 1/a and B/C = 1/b. Lines that pass through the origin or which are horizontal or vertical violate the nonzero condition on a or b and cannot be represented in this form. 6.2. TWO VARIABLES 41

Matrix form

Using the order of the standard form

Ax + By = C,

one can rewrite the equation in matrix form:

( ) ( ) x ( ) AB = C . y

Further, this representation extends to systems of linear equations.

A1x + B1y = C1,

A2x + B2y = C2, becomes:

( )( ) ( ) A B x C 1 1 = 1 . A2 B2 y C2

Since this extends easily to higher dimensions, it is a common representation in linear algebra, and in computer programming. There are named methods for solving system of linear equations, like Gauss-Jordan which can be expressed as matrix elementary row operations.

Parametric form

x = T t + U and

y = V t + W.

Two simultaneous equations in terms of a variable parameter t, with slope m = V / T, x-intercept (VU - WT)/ V and y-intercept (WT - VU)/ T. This can also be related to the two-point form, where T = p - h, U = h, V = q - k, and W = k: x = (p − h)t + h and y = (q − k)t + k.

In this case t varies from 0 at point (h,k) to 1 at point (p,q), with values of t between 0 and 1 providing interpolation and other values of t providing extrapolation.

2D vector determinant form

The equation of a line can also be written as the determinant of two vectors. If P1 and P2 are unique points on the line, then P will also be a point on the line if the following is true: 42 CHAPTER 6. LINEAR EQUATION

−−→ −−−→ det(P1P, P1P2) = 0.

One way to understand this formula is to use the fact that the determinant of two vectors on the plane will give the area of the parallelogram they form. Therefore, if the determinant equals zero then the parallelogram has no area, and that will happen when two vectors are on the same line. −−→ To expand on this we can say that P = (x , y ) , P = (x , y ) and P = (x, y) . Thus P P = (x − x , y − y ) −−−→ 1 1 1 2 2 2 1 1 1 and P1P2 = (x2 − x1, y2 − y1) , then the above equation becomes:

( ) x − x y − y det 1 1 = 0. x2 − x1 y2 − y1

Thus,

(x − x1)(y2 − y1) − (y − y1)(x2 − x1) = 0.

Ergo,

(x − x1)(y2 − y1) = (y − y1)(x2 − x1).

Then dividing both side by (x2 − x1) would result in the “Two-point form” shown above, but leaving it here allows the equation to still be valid when x1 = x2 .

Special cases

y = b y 3 2 1

-3 -2 -10 1 2 3 x -1 -2 y=b -3 (0,b) -4

Horizontal Line y = b 6.2. TWO VARIABLES 43

This is a special case of the standard form where A = 0 and B = 1, or of the slope-intercept form where the slope m = 0. The graph is a horizontal line with y-intercept equal to b. There is no x-intercept, unless b = 0, in which case the graph of the line is the x-axis, and so every real number is an x-intercept.

x = a y 3 2 1 (a,0)

-3 -2 -10 1 2 3 x -1 -2 -3 -4 x=a

Vertical Line x = a

This is a special case of the standard form where A = 1 and B = 0. The graph is a vertical line with x-intercept equal to a. The slope is undeﬁned. There is no y-intercept, unless a = 0, in which case the graph of the line is the y-axis, and so every real number is a y-intercept. This is the only type of line which is not the graph of a function (it obviously fails the vertical line test).

6.2.2 Connection with linear functions

A linear equation, written in the form y = f(x) whose graph crosses the origin (x,y) = (0,0), that is, whose y-intercept is 0, has the following properties:

f(x1 + x2) = f(x1) + f(x2)

and

f(ax) = af(x),

where a is any scalar. A function which satisﬁes these properties is called a linear function (or linear operator, or more generally a linear map). However, linear equations that have non-zero y-intercepts, when written in this manner, produce functions which will have neither property above and hence are not linear functions in this sense. They are known as aﬃne functions. 44 CHAPTER 6. LINEAR EQUATION

6.2.3 Examples

An everyday example of the use of diﬀerent forms of linear equations is computation of tax with tax brackets. This is commonly done using either point–slope form or slope–intercept form; see Progressive tax#Computation for details.

6.3 More than two variables

A linear equation can involve more than two variables. Every linear equation in n unknowns may be rewritten

a1x1 + a2x2 + ··· + anxn = b,

where, a1, a2, ..., an represent numbers, called the coeﬃcients, x1, x2, ..., xn are the unknowns, and b is called the constant term. When dealing with three or fewer variables, it is common to use x, y and z instead of x1, x2 and x3. If all the coeﬃcients are zero, then either b ≠ 0 and the equation does not have any solution, or b = 0 and every set of values for the unknowns is a solution.

If at least one coeﬃcient is nonzero, a permutation of the subscripts allows to suppose a1 ≠ 0, and rewrite the equation

a2 an x1 = b − x2 − · · · − xn. a1 a1

In other words, if ai ≠ 0, one may choose arbitrary values for all the unknowns except xi, and express xi in term of these values. If n = 3 the set of the solutions is a plane in a three-dimensional space. More generally, the set of the solutions is an (n – 1)-dimensional hyperplane in a n-dimensional Euclidean space (or affine space if the coefficients are complex numbers or belong to any field).

6.4 See also

• Line (geometry)

• System of linear equations

• Linear equation over a ring

• Algebraic equation

• Linear belief function

• Linear inequality

6.5 Notes

[1] Barnett, Ziegler & Byleen 2008, pg. 15

6.6 References

• Barnett, R.A.; Ziegler, M.R.; Byleen, K.E. (2008), College Mathematics for Business, Economics, Life Sciences and the Social Sciences (11th ed.), Upper Saddle River, N.J.: Pearson, ISBN 0-13-157225-3 6.7. EXTERNAL LINKS 45

6.7 External links

• Linear Equations and Inequalities Open Elementary Algebra textbook chapter on linear equations and inequalities. • Hazewinkel, Michiel, ed. (2001), “Linear equation”, Encyclopedia of Mathematics, Springer, ISBN 978-1- 55608-010-4 Chapter 7

Mathematics

This article is about the study of topics such as quantity and structure. For other uses, see Mathematics (disambiguation). “Math” redirects here. For other uses, see Math (disambiguation).

Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.[1]

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers),[2] structure,[3] space,[2] and change.[4][5][6] There is a range of views among mathematicians and philosophers as to the exact scope and deﬁnition of mathematics.[7][8] Mathematicians seek out patterns[9][10] and use them to formulate new conjectures. Mathematicians resolve the truth

46 7.1. HISTORY 47

or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.[11] Galileo Galilei (1564–1642) said, “The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth.”[12] Carl Friedrich Gauss (1777–1855) referred to mathematics as “the Queen of the Sciences”.[13] Benjamin Peirce (1809–1880) called mathematics “the science that draws necessary conclusions”.[14] David Hilbert said of mathematics: “We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.”[15] Albert Einstein (1879–1955) stated that “as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”[16] French mathematician Claire Voisin states “There is creative drive in mathematics, it’s all about movement trying to express itself.” [17] Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries, which has led to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.[18]

7.1 History

7.1.1 Evolution

Main article: History of mathematics

The evolution of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, which is shared by many animals,[19] was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely quantity of their members. As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also recognized how to count abstract quantities, like time – days, seasons, years.[20] More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy.[21] The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time. In Babylonian mathematics elementary arithmetic (addition, subtraction, multiplication and division) first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus. Between 600 and 300 BC the Ancient Greeks began a systematic study of mathematics in its own right with Greek mathematics.[22] Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, “The number of papers and 48 CHAPTER 7. MATHEMATICS

Greek mathematician Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the Pythagorean theorem

books included in the Mathematical Reviews database since 1940 (the ﬁrst year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs.”[23] 7.1. HISTORY 49 0 1 2 3 4

5 6 7 8 9

10 11 12 13 14

15 16 17 18 19

Mayan numerals

7.1.2 Etymology

The word mathematics comes from the Greek μάθημα (máthēma), which, in the ancient Greek language, means “that which is learnt”,[24] “what one gets to know”, hence also “study” and “science”, and in modern Greek just “lesson”. The word máthēma is derived from μανθάνω (manthano), while the modern Greek equivalent is μαθαίνω (mathaino), both of which mean “to learn”. In Greece, the word for “mathematics” came to have the narrower and more technical meaning “mathematical study” even in Classical times.[25] Its adjective is μαθηματικός (mathēmatikós), meaning “related to learning” or “studious”, which likewise further came to mean “mathematical”. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: ars mathematica, meant “the mathematical art”. In Latin, and in English until around 1700, the term mathematics more commonly meant “astrology” (or sometimes “astronomy”) rather than “mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations: a particularly notorious one is Saint Augustine's warning that Chris- tians should beware of mathematici meaning astrologers, which is sometimes mistranslated as a condemnation of 50 CHAPTER 7. MATHEMATICS

mathematicians.[26] The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle (384–322 BC), and meaning roughly “all things mathematical"; although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from the Greek.[27] In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in English-speaking North America, math.[28]

7.2 Deﬁnitions of mathematics

Main article: Definitions of mathematics Aristotle defined mathematics as “the science of quantity”, and this definition prevailed until the 18th century.[29] Starting in the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions.[30] Some of these definitions emphasize the deductive character of much of mathematics, some emphasize its abstractness, some emphasize certain topics within mathematics. Today, no consensus on the definition of mathematics prevails, even among professionals.[7] There is not even consensus on whether mathematics is an art or a science.[8] A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.[7] Some just say, “Mathematics is what mathematicians do.”[7] Three leading types of definition of mathematics are called logicist, intuitionist, and formalist, each reflecting a different philosophical school of thought.[31] All have severe problems, none has widespread acceptance, and no reconciliation seems possible.[31] An early definition of mathematics in terms of logic was Benjamin Peirce's “the science that draws necessary conclusions” (1870).[32] In the Principia Mathematica, Bertrand Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proven entirely in terms of symbolic logic. A logicist definition of mathematics is Russell’s “All Mathematics is Symbolic Logic” (1903).[33] Intuitionist definitions, developing from the philosophy of mathematician L.E.J. Brouwer, identify mathematics with certain mental phenomena. An example of an intuitionist definition is “Mathematics is the mental activity which consists in carrying out constructs one after the other.”[31] A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid according to other definitions. In particular, while other philosophies of mathematics allow objects that can be proven to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct. Formalist definitions identify mathematics with its symbols and the rules for operating on them. Haskell Curry defined mathematics simply as “the science of formal systems”.[34] A formal system is a set of symbols, or tokens, and some rules telling how the tokens may be combined into formulas. In formal systems, the word axiom has a special meaning, different from the ordinary meaning of “a self-evident truth”. In formal systems, an axiom is a combination of tokens that is included in a given formal system without needing to be derived using the rules of the system.

7.2.1 Mathematics as science

Gauss referred to mathematics as “the Queen of the Sciences”.[13] In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means a “field of knowledge”, and this was the original meaning of “science” in English, also; mathematics is in this sense a field of knowledge. The specialization restricting the meaning of “science” to natural science follows the rise of Baconian science, which contrasted “natural science” to scholasticism, the Aristotelean method of inquiring from first principles. The role of empirical experimentation and observation is negligible in mathematics, compared to natural sciences such as psychology, biology, or physics. Albert Einstein stated that “as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”[16] More recently, Marcus du Sautoy has called mathematics “the Queen of Science ... the main driving force behind scientific discovery”.[35] Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the 7.2. DEFINITIONS OF MATHEMATICS 51

Leonardo Fibonacci, the Italian mathematician who established the Hindu–Arabic numeral system to the Western World

definition of Karl Popper.[36] However, in the 1930s Gödel’s incompleteness theorems convinced many mathematicians that mathematics cannot be reduced to logic alone, and Karl Popper concluded that “most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently.”[37] Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are 52 CHAPTER 7. MATHEMATICS

Carl Friedrich Gauss, known as the prince of mathematicians intended to correspond to reality. The theoretical physicist J.M. Ziman proposed that science is public knowledge, and thus includes mathematics.[38] Mathematics shares much in common with many ﬁelds in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics. The opinions of mathematicians on this matter are varied. Many mathematicians feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others feel that 7.3. INSPIRATION, PURE AND APPLIED MATHEMATICS, AND AESTHETICS 53

to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics.

7.3 Inspiration, pure and applied mathematics, and aesthetics

Main article: Mathematical beauty

Isaac Newton (left) and Gottfried Wilhelm Leibniz (right), developers of inﬁnitesimal calculus

Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement, architecture and later astronomy; today, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today’s string theory, a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics.[39] Some mathematics is relevant only in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. A distinction is often made between pure mathematics and applied mathematics. However pure mathematics topics often turn out to have applications, e.g. number theory in cryptography. This remarkable fact, that even the “purest” mathematics often turns out to have practical applications, is what Eugene Wigner has called "the unreason- 54 CHAPTER 7. MATHEMATICS

able effectiveness of mathematics".[40] As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest Mathematics Subject Classification runs to 46 pages.[41] Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science. For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant proof, such as Euclid's proof that there are infinitely many prime numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. G.H. Hardy in A Mathematician’s Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He identified criteria such as significance, unexpectedness, in- evitability, and economy as factors that contribute to a mathematical aesthetic.[42] Mathematicians often strive to find proofs that are particularly elegant, proofs from “The Book” of God according to Paul Erdős.[43][44] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.

7.4 Notation, language, and rigor

Main article: Mathematical notation Most of the mathematical notation in use today was not invented until the 16th century.[45] Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery.[46] Euler (1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict syntax (which to a limited extent varies from author to author and from discipline to discipline) and encodes information that would be difficult to write in any other way. Mathematical language can be difficult to understand for beginners. Words such as or and only have more precise meanings than in everyday speech. Moreover, words such as open and field have been given specialized mathematical meanings. Technical terms such as homeomorphism and integrable have precise meanings in mathematics. Addition- ally, shorthand phrases such as iff for "if and only if" belong to mathematical jargon. There is a reason for special notation and technical vocabulary: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as “rigor”. Mathematical proof is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject.[47] The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.[48] Axioms in traditional thought were “self-evident truths”, but that conception is problematic.[49] At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert’s program to put all of mathematics on a firm axiomatic basis, but according to Gödel’s incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is impossible. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.[50]

7.5 Fields of mathematics

See also: Areas of mathematics and Glossary of areas of mathematics Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other ﬁelds: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty. 7.5. FIELDS OF MATHEMATICS 55

Leonhard Euler, who created and popularized much of the mathematical notation used today

7.5.1 Foundations and philosophy

In order to clarify the foundations of mathematics, the ﬁelds of mathematical logic and set theory were developed. Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects. Category theory, which deals in an abstract way with mathematical structures and relationships between them, is still in development. The phrase “crisis of foundations” describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930.[51] Some disagreement about the foundations of mathematics continues to the present day. The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor’s set theory and the Brouwer–Hilbert controversy. 56 CHAPTER 7. MATHEMATICS

An abacus, a simple calculating tool used since ancient times

Mathematical logic is concerned with setting mathematics within a rigorous axiomatic framework, and studying the implications of such a framework. As such, it is home to Gödel’s incompleteness theorems which (informally) imply that any eﬀective formal system that contains basic arithmetic, if sound (meaning that all theorems that can be proven are true), is necessarily incomplete (meaning that there are true theorems which cannot be proved in that system). Whatever ﬁnite collection of number-theoretical axioms is taken as a foundation, Gödel showed how to construct a formal statement that is a true number-theoretical fact, but which does not follow from those axioms. Therefore, no formal system is a complete axiomatization of full number theory. Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer science, as well as to category theory. Theoretical computer science includes computability theory, computational complexity theory, and information theory. Computability theory examines the limitations of various theoretical models of the computer, including the most well-known model – the Turing machine. Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with the rapid advancement of computer hardware. A famous problem is the "P = NP?" problem, one of the Millennium Prize Problems.[52] Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence deals with concepts such as compression and entropy.

7.5.2 Pure mathematics

Quantity

The study of quantity starts with numbers, first the familiar natural numbers and integers (“whole numbers”) and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, from which come such popular results as Fermat’s Last Theorem. The twin prime conjecture and Goldbach’s conjecture are two unsolved problems in number theory. As the number system is further developed, the integers are recognized as a subset of the rational numbers ("fractions"). These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalized to complex numbers. These are the first steps of a hierarchy of numbers that goes on to include quaternions and octonions. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of "infinity". Another area of study is size, which leads to the cardinal numbers and then to another conception of infinity: the aleph numbers, which allow meaningful comparison of the size of infinitely large sets. 7.5. FIELDS OF MATHEMATICS 57

Structure

Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory studies properties of the set of integers that can be expressed in terms of arithmetic operations. Moreover, it frequently happens that different such structured sets (or structures) exhibit similar properties, which makes it possible, by a further step of abstraction, to state axioms for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study groups, rings, fields and other abstract systems; together such studies (for structures defined by algebraic operations) constitute the domain of abstract algebra. By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using Galois theory, which involves field theory and group theory. Another example of an algebraic theory is linear algebra, which is the general study of vector spaces, whose elements called vectors have both quantity and direction, and can be used to model (relations between) points in space. This is one example of the phenomenon that the originally unrelated areas of geometry and algebra have very strong interactions in modern mathematics. Combinatorics studies ways of enumerating the number of objects that fit a given structure.

Space

The study of space originates with geometry – in particular, Euclidean geometry. Trigonometry is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions; it combines space and numbers, and encompasses the well-known Pythagorean theorem. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Convex and discrete geometry were developed to solve problems in number theory and functional analysis but now are pursued with an eye on applications in optimization and computer science. Within differential geometry are the concepts of fiber bundles and calculus on manifolds, in particular, vector and tensor calculus. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may have been the greatest growth area in 20th-century mathematics; it includes point-set topology, set-theoretic topology, algebraic topology and differential topology. In particular, instances of modern day topology are metrizability theory, axiomatic set theory, homotopy theory, and Morse theory. Topology also includes the now solved Poincaré conjecture, and the still unsolved areas of the Hodge conjecture. Other results in geometry and topology, including the four color theorem and Kepler conjecture, have been proved only with the help of computers.

Change

Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and functions of a real variable is known as real analysis, with complex analysis the equivalent field for the complex numbers. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. 58 CHAPTER 7. MATHEMATICS

7.5.3 Applied mathematics

Applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, “applied mathematics” is a mathematical science with specialized knowledge. The term applied mathematics also describes the professional specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the “formulation, study, and use of mathematical models” in science, engineering, and other areas of mathematical practice. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics.

Statistics and other decision sciences

Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. Statisticians (working as part of a research project) “create data that makes sense” with random sampling and with randomized experiments;[53] the design of a statistical sample or experiment specifies the analysis of the data (before the data be available). When reconsidering data from experiments and samples or when analyzing data from observational studies, statisticians “make sense of the data” using the art of modelling and the theory of inference – with model selection and estimation; the estimated models and consequential predictions should be tested on new data.[54] Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints: For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.[55] Because of its use of optimization, the mathematical theory of statistics shares concerns with other decision sciences, such as operations research, control theory, and mathematical economics.[56]

Computational mathematics

Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis includes the study of approximation and discretization broadly with special concern for rounding errors. Numerical analysis and, more broadly, scientiﬁc computing also study non- analytic topics of mathematical science, especially algorithmic matrix and graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.

7.6 Mathematical awards

Arguably the most prestigious award in mathematics is the Fields Medal,[57][58] established in 1936 and now awarded every four years. The Fields Medal is often considered a mathematical equivalent to the Nobel Prize. The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize, was introduced in 2003. The Chern Medal was introduced in 2010 to recognize lifetime achievement. These accolades are awarded in recognition of a particular body of work, which may be innovational, or provide a solution to an outstanding problem in an established ﬁeld. A famous list of 23 open problems, called "Hilbert’s problems", was compiled in 1900 by German mathematician David Hilbert. This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. A solution to each of these problems carries a $1 million reward, and only one (the Riemann hypothesis) is duplicated in Hilbert’s problems. 7.7. SEE ALSO 59

7.7 See also

Main article: Lists of mathematics topics

• Mathematics and art

• Mathematics education

• Relationship between mathematics and physics

• STEM ﬁelds

7.8 Notes

[1] No likeness or description of Euclid’s physical appearance made during his lifetime survived antiquity. Therefore, Euclid’s depiction in works of art depends on the artist’s imagination (see Euclid).

[2] “mathematics, n.". Oxford English Dictionary. Oxford University Press. 2012. Retrieved June 16, 2012. The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis.

[3] Kneebone, G.T. (1963). Mathematical Logic and the Foundations of Mathematics: An Introductory Survey. Dover. pp. 4. ISBN 0-486-41712-3. Mathematics ... is simply the study of abstract structures, or formal patterns of connectedness.

[4] LaTorre, Donald R., John W. Kenelly, Iris B. Reed, Laurel R. Carpenter, and Cynthia R Harris (2011). Calculus Concepts: An Informal Approach to the Mathematics of Change. Cengage Learning. pp. 2. ISBN 1-4390-4957-2. Calculus is the study of change—how things change, and how quickly they change.

[5] Ramana (2007). Applied Mathematics. Tata McGraw–Hill Education. p. 2.10. ISBN 0-07-066753-5. The mathematical study of change, motion, growth or decay is calculus.

[6] Ziegler, Günter M. (2011). “What Is Mathematics?". An Invitation to Mathematics: From Competitions to Research. Springer. pp. 7. ISBN 3-642-19532-6.

[7] Mura, Roberta (Dec 1993). “Images of Mathematics Held by University Teachers of Mathematical Sciences”. Educational Studies in Mathematics 25 (4): 375–385.

[8] Tobies, Renate and Helmut Neunzert (2012). Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry. Springer. pp. 9. ISBN 3-0348-0229-3. It is ﬁrst necessary to ask what is meant by mathematics in general. Illustrious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form.

[9] Steen, L.A. (April 29, 1988). The Science of Patterns Science, 240: 611–616. And summarized at Association for Super- vision and Curriculum Development, www.ascd.org.

[10] Devlin, Keith, Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientiﬁc American Paperback Library) 1996, ISBN 978-0-7167-5047-5

[11] Eves

[12] Marcus du Sautoy, A Brief History of Mathematics: 1. Newton and Leibniz, BBC Radio 4, September 27, 2010.

[13] Waltershausen

[14] Peirce, p. 97.

[15] Hilbert, D. (1919–20), Natur und Mathematisches Erkennen: Vorlesungen, gehalten 1919–1920 in Göttingen. Nach der Ausarbeitung von Paul Bernays (Edited and with an English introduction by David E. Rowe), Basel, Birkhäuser (1992).

[16] Einstein, p. 28. The quote is Einstein’s answer to the question: “how can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?" He, too, is concerned with The Unreasonable Eﬀectiveness of Mathematics in the Natural Sciences.

[17] “Claire Voisin, Artist of the Abstract”. .cnrs.fr. Retrieved October 13, 2013. 60 CHAPTER 7. MATHEMATICS

[18] Peterson

[19] Dehaene, Stanislas; Dehaene-Lambertz, Ghislaine; Cohen, Laurent (Aug 1998). “Abstract representations of numbers in the animal and human brain”. Trends in Neuroscience 21 (8): 355–361. doi:10.1016/S0166-2236(98)01263-6. PMID 9720604.

[20] See, for example, Raymond L. Wilder, Evolution of Mathematical Concepts; an Elementary Study, passim

[21] Kline 1990, Chapter 1.

[22] "A History of Greek Mathematics: From Thales to Euclid". Thomas Little Heath (1981). ISBN 0-486-24073-8

[23] Sevryuk 2006, pp. 101–109.

[24] “mathematic”. Online Etymology Dictionary.

[25] Both senses can be found in Plato. μαθηματική. Liddell, Henry George; Scott, Robert; A Greek–English Lexicon at the Perseus Project

[26] Cipra, Barry (1982). “St. Augustine v. The Mathematicians”. osu.edu. Ohio State University Mathematics department. Retrieved July 14, 2014.

[27] The Oxford Dictionary of English Etymology, Oxford English Dictionary, sub “mathematics”, “mathematic”, “mathematics”

[28] “maths, n." and “math, n.3". Oxford English Dictionary, on-line version (2012).

[29] James Franklin, “Aristotelian Realism” in Philosophy of Mathematics”, ed. A.D. Irvine, p. 104. Elsevier (2009).

[30] Cajori, Florian (1893). A History of Mathematics. American Mathematical Society (1991 reprint). pp. 285–6. ISBN 0-8218-2102-4.

[31] Snapper, Ernst (September 1979). “The Three Crises in Mathematics: Logicism, Intuitionism, and Formalism”. Mathe- matics Magazine 52 (4): 207–16. doi:10.2307/2689412. JSTOR 2689412.

[32] Peirce, Benjamin (1882). Linear Associative Algebra. p. 1.

[33] Bertrand Russell, The Principles of Mathematics, p. 5. University Press, Cambridge (1903)

[34] Curry, Haskell (1951). Outlines of a Formalist Philosophy of Mathematics. Elsevier. pp. 56. ISBN 0-444-53368-0.

[35] Marcus du Sautoy, A Brief History of Mathematics: 10. Nicolas Bourbaki, BBC Radio 4, October 1, 2010.

[36] Shasha, Dennis Elliot; Lazere, Cathy A. (1998). Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists. Springer. p. 228.

[37] Popper 1995, p. 56

[38] Ziman

[39] Johnson, Gerald W.; Lapidus, Michel L. (2002). The Feynman Integral and Feynman’s Operational Calculus. Oxford University Press. ISBN 0-8218-2413-9.

[40] Wigner, Eugene (1960). “The Unreasonable Eﬀectiveness of Mathematics in the Natural Sciences”. Communications on Pure and Applied Mathematics 13 (1): 1–14. doi:10.1002/cpa.3160130102.

[41] “Mathematics Subject Classiﬁcation 2010” (PDF). Retrieved November 9, 2010.

[42] Hardy, G.H. (1940). A Mathematician’s Apology. Cambridge University Press. ISBN 0-521-42706-1.

[43] Gold, Bonnie; Simons, Rogers A. (2008). Proof and Other Dilemmas: Mathematics and Philosophy. MAA.

[44] Aigner, Martin; Ziegler, Günter M. (2001). Proofs from The Book. Springer. ISBN 3-540-40460-0.

[45] “Earliest Uses of Various Mathematical Symbols”. Retrieved September 14, 2014.

[46] Kline, p. 140, on Diophantus; p. 261, on Vieta.

[47] See false proof for simple examples of what can go wrong in a formal proof.

[48] Ivars Peterson, The Mathematical Tourist, Freeman, 1988, ISBN 0-7167-1953-3. p. 4 “A few complain that the computer program can't be veriﬁed properly”, (in reference to the Haken–Apple proof of the Four Color Theorem). 7.9. REFERENCES 61

[49] " The method of “postulating” what we want has many advantages; they are the same as the advantages of theft over honest toil.” Bertrand Russell (1919), Introduction to Mathematical Philosophy, New York and London, p 71.

[50] Patrick Suppes, Axiomatic Set Theory, Dover, 1972, ISBN 0-486-61630-4. p. 1, “Among the many branches of modern mathematics set theory occupies a unique place: with a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects.”

[51] Luke Howard Hodgkin & Luke Hodgkin, A History of Mathematics, Oxford University Press, 2005.

[52] Clay Mathematics Institute, P=NP, claymath.org

[53] Rao, C.R. (1997) Statistics and Truth: Putting Chance to Work, World Scientiﬁc. ISBN 981-02-3111-3

[54] Like other mathematical sciences such as physics and computer science, statistics is an autonomous discipline rather than a branch of applied mathematics. Like research physicists and computer scientists, research statisticians are mathematical scientists. Many statisticians have a degree in mathematics, and some statisticians are also mathematicians.

[55] Rao, C.R. (1981). “Foreword”. In Arthanari, T.S.; Dodge, Yadolah. Mathematical programming in statistics. Wiley Series in Probability and Mathematical Statistics. New York: Wiley. pp. vii–viii. ISBN 0-471-08073-X. MR 607328.

[56] Whittle (1994, pp. 10–11 and 14–18): Whittle, Peter (1994). “Almost home”. In Kelly, F.P. Probability, statistics and optimisation: A Tribute to Peter Whittle (previously “A realised path: The Cambridge Statistical Laboratory upto 1993 (revised 2002)" ed.). Chichester: John Wiley. pp. 1–28. ISBN 0-471-94829-2.

[57] Monastyrsky 2001:"The Fields Medal is now indisputably the best known and most inﬂuential award in mathematics."

[58] Riehm 2002, pp. 778–782.

7.9 References

• Courant, Richard and H. Robbins, What Is Mathematics? : An Elementary Approach to Ideas and Methods, Oxford University Press, USA; 2 edition (July 18, 1996). ISBN 0-19-510519-2.

• Einstein, Albert (1923). Sidelights on Relativity: I. Ether and relativity. II. Geometry and experience (translated by G.B. Jeﬀery, D.Sc., and W. Perrett, Ph.D). E.P. Dutton & Co., New York.

• du Sautoy, Marcus, A Brief History of Mathematics, BBC Radio 4 (2010).

• Eves, Howard, An Introduction to the History of Mathematics, Sixth Edition, Saunders, 1990, ISBN 0-03- 029558-0.

• Kline, Morris, Mathematical Thought from Ancient to Modern Times, Oxford University Press, USA; Paperback edition (March 1, 1990). ISBN 0-19-506135-7.

• Monastyrsky, Michael (2001). “Some Trends in Modern Mathematics and the Fields Medal” (PDF). Canadian Mathematical Society. Retrieved July 28, 2006.

• Oxford English Dictionary, second edition, ed. John Simpson and Edmund Weiner, Clarendon Press, 1989, ISBN 0-19-861186-2.

• The Oxford Dictionary of English Etymology, 1983 reprint. ISBN 0-19-861112-9.

• Pappas, Theoni, The Joy Of Mathematics, Wide World Publishing; Revised edition (June 1989). ISBN 0- 933174-65-9.

• Peirce, Benjamin (1881). Peirce, Charles Sanders, ed. “Linear associative algebra”. American Journal of Mathematics (Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C.S. Peirce, of the 1872 lithograph ed.) (Johns Hopkins University) 4 (1–4): 97–229. doi:10.2307/2369153. JSTOR 2369153. Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C. S. Peirce, of the 1872 lithograph ed. Google Eprint and as an extract, D. Van Nostrand, 1882, Google Eprint..

• Peterson, Ivars, Mathematical Tourist, New and Updated Snapshots of Modern Mathematics, Owl Books, 2001, ISBN 0-8050-7159-8. 62 CHAPTER 7. MATHEMATICS

• Popper, Karl R. (1995). “On knowledge”. In Search of a Better World: Lectures and Essays from Thirty Years. Routledge. ISBN 0-415-13548-6. • Riehm, Carl (August 2002). “The Early History of the Fields Medal” (PDF). Notices of the AMS (AMS) 49 (7): 778–782. • Sevryuk, Mikhail B. (January 2006). “Book Reviews” (PDF). Bulletin of the American Mathematical Society 43 (1): 101–109. doi:10.1090/S0273-0979-05-01069-4. Retrieved June 24, 2006. • Waltershausen, Wolfgang Sartorius von (1965) [ﬁrst published 1856]. Gauss zum Gedächtniss. Sändig Reprint Verlag H. R. Wohlwend. ASIN B0000BN5SQ. ISBN 3-253-01702-8. ASIN 3253017028.

7.10 Further reading

• Benson, Donald C., The Moment of Proof: Mathematical Epiphanies, Oxford University Press, USA; New Ed edition (December 14, 2000). ISBN 0-19-513919-4. • Boyer, Carl B., A History of Mathematics, Wiley; 2nd edition, revised by Uta C. Merzbach, (March 6, 1991). ISBN 0-471-54397-7.—A concise history of mathematics from the Concept of Number to contemporary Mathematics. • Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Mariner Books; Reprint edition (January 14, 1999). ISBN 0-395-92968-7. • Gullberg, Jan, Mathematics – From the Birth of Numbers. W. W. Norton & Company; 1st edition (October 1997). ISBN 0-393-04002-X. • Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. – A translated and expanded version of a Soviet mathematics encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM, and online. • Jourdain, Philip E. B., The Nature of Mathematics, in The World of Mathematics, James R. Newman, editor, Dover Publications, 2003, ISBN 0-486-43268-8. • Maier, Annaliese, At the Threshold of Exact Science: Selected Writings of Annaliese Maier on Late Medieval Natural Philosophy, edited by Steven Sargent, Philadelphia: University of Pennsylvania Press, 1982.

7.11 External links

• Mathematics at Encyclopædia Britannica • Mathematics on In Our Time at the BBC.(listen now) • Free Mathematics books Free Mathematics books collection. • Encyclopaedia of Mathematics online encyclopaedia from Springer, Graduate-level reference work with over 8,000 entries, illuminating nearly 50,000 notions in mathematics. • HyperMath site at Georgia State University • FreeScience Library The mathematics section of FreeScience library • Rusin, Dave: The Mathematical Atlas. A guided tour through the various branches of modern mathematics. (Can also be found at NIU.edu.) • Polyanin, Andrei: EqWorld: The World of Mathematical Equations. An online resource focusing on algebraic, ordinary diﬀerential, partial diﬀerential (mathematical physics), integral, and other mathematical equations. • Cain, George: Online Mathematics Textbooks available free online. • Tricki, Wiki-style site that is intended to develop into a large store of useful mathematical problem-solving techniques. 7.11. EXTERNAL LINKS 63

• Mathematical Structures, list information about classes of mathematical structures.

• Mathematician Biographies. The MacTutor History of Mathematics archive Extensive history and quotes from all famous mathematicians.

• Metamath. A site and a language, that formalize mathematics from its foundations. • Nrich, a prize-winning site for students from age ﬁve from Cambridge University

• Open Problem Garden, a wiki of open problems in mathematics • Planet Math. An online mathematics encyclopedia under construction, focusing on modern mathematics. Uses the Attribution-ShareAlike license, allowing article exchange with Wikipedia. Uses TeX markup. • Some mathematics applets, at MIT

• Weisstein, Eric et al.: MathWorld: World of Mathematics. An online encyclopedia of mathematics.

• Patrick Jones’ Video Tutorials on Mathematics • Citizendium: Theory (mathematics).

• du Sautoy, Marcus, A Brief History of Mathematics, BBC Radio 4 (2010). • MathOverﬂow A Q&A site for research-level mathematics

• Math – Khan Academy • National Museum of Mathematics, located in New York City Chapter 8

Matrix (mathematics)

For other uses, see Matrix. “Matrix theory” redirects here. For the physics topic, see Matrix string theory. In mathematics, a matrix (plural matrices) is a rectangular array[1]—of numbers, symbols, or expressions, arranged

ai,j n columns j changes m rows . . a1,1 a1,2 a1,3 . i c . . a2,1 . h a2,2 a2,3 a n . . a3,1 a3,2 a3,3 . g e . . . . s ......

Each element of a matrix is often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and ﬁrst column of a matrix A. in rows and columns[2][3]—that is treated in certain prescribed ways. One such way is to state the order of the matrix. For example, the order of the matrix below is a 2x3 matrix because there are two rows and three columns. The individual items in a matrix are called its elements or entries.[4]

64 8.1. DEFINITION 65

[ ] 1 9 −13 . 20 5 −6

Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix’s eigenvalues and eigenvectors. Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions. A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function.

8.1 Deﬁnition

A matrix is a rectangular array of numbers or other mathematical objects, for which operations such as addition and multiplication are deﬁned.[6] Most commonly, a matrix over a ﬁeld F is a rectangular array of scalars from F.[7][8] Most of this article focuses on real and complex matrices, i.e., matrices whose elements are real numbers or complex numbers, respectively. More general types of entries are discussed below. For instance, this is a real matrix:

  −1.3 0.6 A =  20.4 5.5 . 9.7 −6.2

The numbers, symbols or expressions in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.

8.1.1 Size

The size of a matrix is defined by the number of rows and columns that it contains. A matrix with m rows and n columns is called an m × n matrix or m-by-n matrix, while m and n are called its dimensions. For example, the matrix A above is a 3 × 2 matrix. Matrices which have a single row are called row vectors, and those which have a single column are called column vectors. A matrix which has the same number of rows and columns is called a square matrix. A matrix with an infinite number of rows or columns (or both) is called an infinite matrix. In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an empty matrix. 66 CHAPTER 8. MATRIX (MATHEMATICS)

8.2 Notation

Matrices are commonly written in box brackets or an alternative notation uses large parentheses instead of box brackets:

    a11 a12 ··· a1n a11 a12 ··· a1n     a21 a22 ··· a2n a21 a22 ··· a2n     m×n A =  . . . .  =  . . . .  ∈ R .  . . .. .   . . .. .  am1 am2 ··· amn am1 am2 ··· amn

The speciﬁcs of symbolic matrix notation varies widely, with some prevailing trends. Matrices are usually symbolized using upper-case letters (such as A in the examples above), while the corresponding lower-case letters, with two subscript indices (e.g., a11, or a₁,₁), represent the entries. In addition to using upper-case letters to symbolize matrices, many authors use a special typographical style, commonly boldface upright (non-italic), to further distinguish matrices from other mathematical objects. An alternative notation involves the use of a double-underline with the variable name, with or without boldface style, (e.g., A ). The entry in the i-th row and j-th column of a matrix A is sometimes referred to as the i,j,(i,j), or (i,j)th entry of the matrix, and most commonly denoted as ai,j, or aij. Alternative notations for that entry are A[i,j] or Ai,j. For example, the (1,3) entry of the following matrix A is 5 (also denoted a13, a₁,₃, A[1,3] or A1,3):

  4 −7 5 0 A = −2 0 11 8  19 1 −3 12

Sometimes, the entries of a matrix can be deﬁned by a formula such as ai,j = f(i, j). For example, each of the entries of the following matrix A is determined by aij = i − j.

  0 −1 −2 −3 A = 1 0 −1 −2 2 1 0 −1

In this case, the matrix itself is sometimes defined by that formula, within square brackets or double parenthesis. For example, the matrix above is defined as A = [i-j], or A = ((i-j)). If matrix size is m × n, the above-mentioned formula f(i, j) is valid for any i = 1, ..., m and any j = 1, ..., n. This can be either specified separately, or using m × n as a subscript. For instance, the matrix A above is 3 × 4 and can be defined as A = [i − j](i = 1, 2, 3; j = 1, ..., 4), or A = [i − j]3×4. Some programming languages utilize doubly subscripted arrays (or arrays of arrays) to represent an m-×-n matrix. Some programming languages start the numbering of array indexes at zero, in which case the entries of an m-by-n matrix are indexed by 0 ≤ i ≤ m − 1 and 0 ≤ j ≤ n − 1.[9] This article follows the more common convention in mathematical writing where enumeration starts from 1. The set of all m-by-n matrices is denoted 필(m, n).

8.3 Basic operations

There are a number of basic operations that can be applied to modify matrices, called matrix addition, scalar multiplication, transposition, matrix multiplication, row operations, and submatrix.[11]

8.3.1 Addition, scalar multiplication and transposition

Main articles: Matrix addition, Scalar multiplication and Transpose 8.3. BASIC OPERATIONS 67

Familiar properties of numbers extend to these operations of matrices: for example, addition is commutative, i.e., the matrix sum does not depend on the order of the summands: A + B = B + A.[12] The transpose is compatible with addition and scalar multiplication, as expressed by (cA)T = c(AT) and (A + B)T = AT + BT. Finally, (AT)T = A.

8.3.2 Matrix multiplication

Main article: Matrix multiplication Multiplication of two matrices is deﬁned if and only if the number of columns of the left matrix is the same as the B

b1,1 b1,2 b1,3

b2,1 b2,2 b2,3

a1,1 a1,2

a a A 2,1 2,2 a3,1 a3,2

a4,1 a4,2

Schematic depiction of the matrix product AB of two matrices A and B.

number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: ∑ ··· n [AB]i,j = Ai,1B1,j + Ai,2B2,j + + Ai,nBn,j = r=1 Ai,rBr,j ,

where 1 ≤ i ≤ m and 1 ≤ j ≤ p.[13] For example, the underlined entry 2340 in the product is calculated as (2 × 1000) + (3 × 100) + (4 × 10) = 2340:

  [ ] 0 1000 [ ] 2 3 4 3 2340 1 100  = . 1 0 0 0 1000 0 10 68 CHAPTER 8. MATRIX (MATHEMATICS)

Matrix multiplication satisfies the rules (AB)C = A(BC)(associativity), and (A+B)C = AC+BC as well as C(A+B) = CA+CB (left and right distributivity), whenever the size of the matrices is such that the various products are defined.[14] The product AB may be defined without BA being defined, namely if A and B are m-by-n and n-by-k matrices, respectively, and m ≠ k. Even if both products are defined, they need not be equal, i.e., generally

AB ≠ BA,

i.e., matrix multiplication is not commutative, in marked contrast to (rational, real, or complex) numbers whose product is independent of the order of the factors. An example of two matrices not commuting with each other is:

[ ][ ] [ ] 1 2 0 1 0 1 = , 3 4 0 0 0 3

whereas

[ ][ ] [ ] 0 1 1 2 3 4 = . 0 0 3 4 0 0

Besides the ordinary matrix multiplication just described, there exist other less frequently used operations on matrices that can be considered forms of multiplication, such as the Hadamard product and the Kronecker product.[15] They arise in solving matrix equations such as the Sylvester equation.

8.3.3 Row operations

Main article: Row operations

There are three types of row operations:

1. row addition, that is adding a row to another.

2. row multiplication, that is multiplying all entries of a row by a non-zero constant;

3. row switching, that is interchanging two rows of a matrix;

These operations are used in a number of ways, including solving linear equations and ﬁnding matrix inverses.

8.3.4 Submatrix

A submatrix of a matrix is obtained by deleting any collection of rows and/or columns.[16][17][18] For example, from the following 3-by-4 matrix, we can construct a 2-by-3 submatrix by removing row 3 and column 2:

  1 2 3 4 [ ] 1 3 4 A = 5 6 7 8  → . 5 7 8 9 10 11 12

The minors and cofactors of a matrix are found by computing the determinant of certain submatrices.[18][19] A principal submatrix is a square submatrix obtained by removing certain rows and columns. The definition varies from author to author. According to some authors, a principal submatrix is a submatrix in which the set of row indices that remain is the same as the set of column indices that remain.[20][21] Other authors define a principal submatrix to be one in which the first k rows and columns, for some number k, are the ones that remain;[22] this type of submatrix has also been called a leading principal submatrix.[23] 8.4. LINEAR EQUATIONS 69

8.4 Linear equations

Main articles: Linear equation and System of linear equations

Matrices can be used to compactly write and work with multiple linear equations, i.e., systems of linear equations. For example, if A is an m-by-n matrix, x designates a column vector (i.e., n×1-matrix) of n variables x1, x2, ..., xn, and b is an m×1-column vector, then the matrix equation

Ax = b

is equivalent to the system of linear equations

A₁,₁x1 + A₁,₂x2 + ... + A₁,nxn = b1 ... [24] Am,₁x1 + Am,₂x2 + ... + Am,nxn = bm .

8.5 Linear transformations

Main articles: Linear transformation and Transformation matrix Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps. A real m-by-n matrix A gives rise to a linear transformation Rn → Rm mapping each vector x in Rn to the (matrix) product Ax, which is a vector in Rm. Conversely, each linear transformation f: Rn → Rm arises from a unique m-by-n matrix A: explicitly, the (i, j)-entry of A is the ith coordinate of f(ej), where ej = (0,...,0,1,0,...,0) is the unit vector with 1 in the jth position and 0 elsewhere. The matrix A is said to represent the linear map f, and A is called the transformation matrix of f. For example, the 2×2 matrix

[ ] a c A = b d can be viewed as the transform of the unit square into a parallelogram with vertices at (0, 0), (a, b), (a + c, b + d), and[ ] ([c, ]d).[ The] parallelogram[ ] pictured at the right is obtained by multiplying A with each of the column vectors 0 1 1 0 , , and in turn. These vectors deﬁne the vertices of the unit square. 0 0 1 1 The following table shows a number of 2-by-2 matrices with the associated linear maps of R2. The blue original is mapped to the green grid and shapes. The origin (0,0) is marked with a black point. Under the 1-to-1 correspondence between matrices and linear maps, matrix multiplication corresponds to composition of maps:[25] if a k-by-m matrix B represents another linear map g : Rm → Rk, then the composition g ∘ f is represented by BA since

(g ∘ f)(x) = g(f(x)) = g(Ax) = B(Ax) = (BA)x.

The last equality follows from the above-mentioned associativity of matrix multiplication. The rank of a matrix A is the maximum number of linearly independent row vectors of the matrix, which is the same as the maximum number of linearly independent column vectors.[26] Equivalently it is the dimension of the image of the linear map represented by A.[27] The rank-nullity theorem states that the dimension of the kernel of a matrix plus the rank equals the number of columns of the matrix.[28]

8.6 Square matrices

Main article: Square matrix 70 CHAPTER 8. MATRIX (MATHEMATICS)

(a+c,b+d)

(c,d)

ad−bc

(a,b)

(0,0)

The vectors represented by a 2-by-2 matrix correspond to the sides of a unit square transformed into a parallelogram.

A square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. The entries aii form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix.

8.6.1 Main types 8.6. SQUARE MATRICES 71

Diagonal and triangular matrices

If all entries of A below the main diagonal are zero, A is called an upper triangular matrix. Similarly if all entries of A above the main diagonal are zero, A is called a lower triangular matrix. If all entries outside the main diagonal are zero, A is called a diagonal matrix.

Identity matrix

The identity matrix In of size n is the n-by-n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g.

  1 0 ··· 0 [ ]   [ ] 0 1 ··· 0 1 0 ··· I1 = 1 ,I2 = , ,In = . . . . 0 1 . . .. . 0 0 ··· 1 It is a square matrix of order n, and also a special kind of diagonal matrix. It is called an identity matrix because multiplication with it leaves a matrix unchanged:

AIn = ImA = A for any m-by-n matrix A.

Symmetric or skew-symmetric matrix

A square matrix A that is equal to its transpose, i.e., A = AT, is a symmetric matrix. If instead, A was equal to the negative of its transpose, i.e., A = −AT, then A is a skew-symmetric matrix. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy A∗ = A, where the star or asterisk denotes the conjugate transpose of the matrix, i.e., the transpose of the complex conjugate of A. By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; i.e., every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real.[29] This theorem can be generalized to inﬁnite-dimensional situations related to matrices with inﬁnitely many rows and columns, see below.

Invertible matrix and its inverse

A square matrix A is called invertible or non-singular if there exists a matrix B such that

AB = BA = In.[30][31]

If B exists, it is unique and is called the inverse matrix of A, denoted A−1.

Deﬁnite matrix

A symmetric n×n-matrix is called positive-definite (respectively negative-definite; indefinite), if for all nonzero vectors x ∈ Rn the associated quadratic form given by

Q(x) = xTAx

takes only positive values (respectively only negative values; both some negative and some positive values).[32] If the quadratic form takes only non-negative (respectively only non-positive) values, the symmetric matrix is called positive-semidefinite (respectively negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite. A symmetric matrix is positive-definite if and only if all its eigenvalues are positive, i.e., the matrix is positive- semidefinite and it is invertible.[33] The table at the right shows two possibilities for 2-by-2 matrices. Allowing as input two different vectors instead yields the bilinear form associated to A: 72 CHAPTER 8. MATRIX (MATHEMATICS)

BA (x, y) = xTAy.[34]

Orthogonal matrix

An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors). Equivalently, a matrix A is orthogonal if its transpose is equal to its inverse:

AT = A−1, which entails

ATA = AAT = I, where I is the identity matrix. An orthogonal matrix A is necessarily invertible (with inverse A−1 = AT), unitary (A−1 = A*), and normal (A*A = AA*). The determinant of any orthogonal matrix is either +1 or −1. A special orthogonal matrix is an orthogonal matrix with determinant +1. As a linear transformation, every orthogonal matrix with determinant +1 is a pure rotation, while every orthogonal matrix with determinant −1 is either a pure reﬂection, or a composition of reﬂection and rotation. The complex analogue of an orthogonal matrix is a unitary matrix.

8.6.2 Main operations

Trace

The trace, tr(A) of a square matrix A is the sum of its diagonal entries. While matrix multiplication is not commutative as mentioned above, the trace of the product of two matrices is independent of the order of the factors:

tr(AB) = tr(BA).

This is immediate from the deﬁnition of matrix multiplication:

∑ ∑ m n tr(AB)= i=1 j=1 Aij Bji=tr(BA). Also, the trace of a matrix is equal to that of its transpose, i.e.,

tr(A) = tr(AT).

Determinant

Main article: Determinant The determinant det(A) or |A| of a square matrix A is a number encoding certain properties of the matrix. A matrix is invertible if and only if its determinant is nonzero. Its absolute value equals the area (in R2) or volume (in R3) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved. The determinant of 2-by-2 matrices is given by

[ ] a b det = ad − bc. c d The determinant of 3-by-3 matrices involves 6 terms (rule of Sarrus). The more lengthy Leibniz formula generalises these two formulae to all dimensions.[35] The determinant of a product of square matrices equals the product of their determinants: 8.7. COMPUTATIONAL ASPECTS 73

0 1 ()1 −1

x2 f(x1 )

f(x2 )

A linear transformation on R2 given by the indicated matrix. The determinant of this matrix is −1, as the area of the green parallelogram at the right is 1, but the map reverses the orientation, since it turns the counterclockwise orientation of the vectors to a clockwise one.

det(AB) = det(A) · det(B).[36]

Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. Interchanging two rows or two columns aﬀects the determinant by multiplying it by −1.[37] Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the Laplace expansion expresses the determinant in terms of minors, i.e., determinants of smaller matrices.[38] This expansion can be used for a recursive deﬁnition of determinants (taking as starting case the determinant of a 1-by-1 matrix, which is its unique entry, or even the determinant of a 0-by-0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve linear systems using Cramer’s rule, where the division of the determinants of two related square matrices equates to the value of each of the system’s variables.[39]

Eigenvalues and eigenvectors

Main article: Eigenvalues and eigenvectors

A number λ and a non-zero vector v satisfying

Av = λv

are called an eigenvalue and an eigenvector of A, respectively.[nb 1][40] The number λ is an eigenvalue of an n×n-matrix A if and only if A−λIn is not invertible, which is equivalent to

det(A − λI) = 0. [41]

The polynomial pA in an indeterminate X given by evaluation the determinant det(XIn−A) is called the characteristic polynomial of A. It is a monic polynomial of degree n. Therefore the polynomial equation pA(λ) = 0 has at most n diﬀerent solutions, i.e., eigenvalues of the matrix.[42] They may be complex even if the entries of A are real. According to the Cayley–Hamilton theorem, pA(A) = 0, that is, the result of substituting the matrix itself into its own characteristic polynomial yields the zero matrix.

8.7 Computational aspects

Matrix calculations can be often performed with diﬀerent techniques. Many problems can be solved by both direct algorithms or iterative approaches. For example, the eigenvectors of a square matrix can be obtained by ﬁnding a 74 CHAPTER 8. MATRIX (MATHEMATICS)

sequence of vectors xn converging to an eigenvector when n tends to infinity.[43] To be able to choose the more appropriate algorithm for each specific problem, it is important to determine both the effectiveness and precision of all the available algorithms. The domain studying these matters is called numerical linear algebra.[44] As with other numerical situations, two main aspects are the complexity of algorithms and their numerical stability. Determining the complexity of an algorithm means finding upper bounds or estimates of how many elementary operations such as additions and multiplications of scalars are necessary to perform some algorithm, e.g., multiplication of matrices. For example, calculating the matrix product of two n-by-n matrix using the definition given above needs n3 multiplications, since for any of the n2 entries of the product, n multiplications are necessary. The Strassen algorithm outperforms this “naive” algorithm; it needs only n2.807 multiplications.[45] A refined approach also incorporates specific features of the computing devices. In many practical situations additional information about the matrices involved is known. An important case are sparse matrices, i.e., matrices most of whose entries are zero. There are specifically adapted algorithms for, say, solving linear systems Ax = b for sparse matrices A, such as the conjugate gradient method.[46] An algorithm is, roughly speaking, numerically stable, if little deviations in the input values do not lead to big deviations in the result. For example, calculating the inverse of a matrix via Laplace’s formula (Adj (A) denotes the adjugate matrix of A)

A−1 = Adj(A) / det(A)

may lead to signiﬁcant rounding errors if the determinant of the matrix is very small. The norm of a matrix can be used to capture the conditioning of linear algebraic problems, such as computing a matrix’s inverse.[47] Although most computer languages are not designed with commands or libraries for matrices, as early as the 1970s, some engineering desktop computers such as the HP 9830 had ROM cartridges to add BASIC commands for matrices. Some computer languages such as APL were designed to manipulate matrices, and various mathematical programs can be used to aid computing with matrices.[48]

8.8 Decomposition

Main articles: Matrix decomposition, Matrix diagonalization, Gaussian elimination and Montante’s method

There are several methods to render matrices into a more easily accessible form. They are generally referred to as matrix decomposition or matrix factorization techniques. The interest of all these techniques is that they preserve certain properties of the matrices in question, such as determinant, rank or inverse, so that these quantities can be calculated after applying the transformation, or that certain matrix operations are algorithmically easier to carry out for some types of matrices. The LU decomposition factors matrices as a product of lower (L) and an upper triangular matrices (U).[49] Once this decomposition is calculated, linear systems can be solved more eﬃciently, by a simple technique called forward and back substitution. Likewise, inverses of triangular matrices are algorithmically easier to calculate. The Gaus- sian elimination is a similar algorithm; it transforms any matrix to row echelon form.[50] Both methods proceed by multiplying the matrix by suitable elementary matrices, which correspond to permuting rows or columns and adding multiples of one row to another row. Singular value decomposition expresses any matrix A as a product UDV∗, where U and V are unitary matrices and D is a diagonal matrix. The eigendecomposition or diagonalization expresses A as a product VDV−1, where D is a diagonal matrix and V is a suitable invertible matrix.[51] If A can be written in this form, it is called diagonalizable. More generally, and applicable to all matrices, the Jordan decomposition transforms a matrix into Jordan normal form, that is to say matrices whose only nonzero entries are the eigenvalues λ1 to λ of A, placed on the main diagonal and possibly entries equal to one directly above the main diagonal, as shown at the right.[52] Given the eigendecomposition, the nth power of A (i.e., n-fold iterated matrix multiplication) can be calculated via

An = (VDV−1)n = VDV−1VDV−1...VDV−1 = VDnV−1

and the power of a diagonal matrix can be calculated by taking the corresponding powers of the diagonal entries, which is much easier than doing the exponentiation for A instead. This can be used to compute the matrix exponential eA, a 8.9. ABSTRACT ALGEBRAIC ASPECTS AND GENERALIZATIONS 75

An example of a matrix in Jordan normal form. The grey blocks are called Jordan blocks. need frequently arising in solving linear diﬀerential equations, matrix logarithms and square roots of matrices.[53] To avoid numerically ill-conditioned situations, further algorithms such as the Schur decomposition can be employed.[54]

8.9 Abstract algebraic aspects and generalizations

Matrices can be generalized in different ways. Abstract algebra uses matrices with entries in more general fields or even rings, while linear algebra codifies properties of matrices in the notion of linear maps. It is possible to consider matrices with infinitely many columns and rows. Another extension are tensors, which can be seen as higher-dimensional arrays of numbers, as opposed to vectors, which can often be realised as sequences of numbers, while matrices are rectangular or two-dimensional arrays of numbers.[55] Matrices, subject to certain requirements tend to form groups known as matrix groups.

8.9.1 Matrices with more general entries

This article focuses on matrices whose entries are real or complex numbers. However, matrices can be considered with much more general types of entries than real or complex numbers. As a first step of generalization, any field, i.e., a set where addition, subtraction, multiplication and division operations are defined and well-behaved, may be used instead of R or C, for example rational numbers or finite fields. For example, coding theory makes use of matrices over finite fields. Wherever eigenvalues are considered, as these are roots of a polynomial they may exist only in a 76 CHAPTER 8. MATRIX (MATHEMATICS) larger field than that of the entries of the matrix; for instance they may be complex in case of a matrix with real entries. The possibility to reinterpret the entries of a matrix as elements of a larger field (e.g., to view a real matrix as a complex matrix whose entries happen to be all real) then allows considering each square matrix to possess a full set of eigenvalues. Alternatively one can consider only matrices with entries in an algebraically closed field, such as C, from the outset. More generally, abstract algebra makes great use of matrices with entries in a ring R.[56] Rings are a more general notion than fields in that a division operation need not exist. The very same addition and multiplication operations of matrices extend to this setting, too. The set M(n, R) of all square n-by-n matrices over R is a ring called matrix ring, isomorphic to the endomorphism ring of the left R-module Rn.[57] If the ring R is commutative, i.e., its multiplication is commutative, then M(n, R) is a unitary noncommutative (unless n = 1) associative algebra over R. The determinant of square matrices over a commutative ring R can still be defined using the Leibniz formula; such a matrix is invertible if and only if its determinant is invertible in R, generalising the situation over a field F, where every nonzero element is invertible.[58] Matrices over superrings are called supermatrices.[59] Matrices do not always have all their entries in the same ring – or even in any ring at all. One special but common case is block matrices, which may be considered as matrices whose entries themselves are matrices. The entries need not be quadratic matrices, and thus need not be members of any ordinary ring; but their sizes must fulfil certain compatibility conditions.

8.9.2 Relationship to linear maps

Linear maps Rn → Rm are equivalent to m-by-n matrices, as described above. More generally, any linear map f: V → W between ﬁnite-dimensional vector spaces can be described by a matrix A = (aij), after choosing bases v1, ..., vn of V, and w1, ..., wm of W (so n is the dimension of V and m is the dimension of W), which is such that

∑m f(vj) = ai,jwi for j = 1, . . . , n. i=1 In other words, column j of A expresses the image of vj in terms of the basis vectors wi of W; thus this relation uniquely determines the entries of the matrix A. Note that the matrix depends on the choice of the bases: different choices of bases give rise to different, but equivalent matrices.[60] Many of the above concrete notions can be reinterpreted in this light, for example, the transpose matrix AT describes the transpose of the linear map given by A, with respect to the dual bases.[61] These properties can be restated in a more natural way: the category of all matrices with entries in a field k with multiplication as composition is equivalent to the category of finite dimensional vector spaces and linear maps over this field. More generally, the set of m×n matrices can be used to represent the R-linear maps between the free modules Rm and Rn for an arbitrary ring R with unity. When n = m composition of these maps is possible, and this gives rise to the matrix ring of n×n matrices representing the endomorphism ring of Rn.

8.9.3 Matrix groups

Main article: Matrix group

A group is a mathematical structure consisting of a set of objects together with a binary operation, i.e., an operation combining any two objects to a third, subject to certain requirements.[62] A group in which the objects are matrices and the group operation is matrix multiplication is called a matrix group.[nb 2][63] Since in a group every element has to be invertible, the most general matrix groups are the groups of all invertible matrices of a given size, called the general linear groups. Any property of matrices that is preserved under matrix products and inverses can be used to deﬁne further matrix groups. For example, matrices with a given size and with a determinant of 1 form a subgroup of (i.e., a smaller group contained in) their general linear group, called a special linear group.[64] Orthogonal matrices, determined by the condition

MTM = I, 8.9. ABSTRACT ALGEBRAIC ASPECTS AND GENERALIZATIONS 77

form the orthogonal group.[65] Every orthogonal matrix has determinant 1 or −1. Orthogonal matrices with determinant 1 form a subgroup called special orthogonal group. Every ﬁnite group is isomorphic to a matrix group, as one can see by considering the regular representation of the symmetric group.[66] General groups can be studied using matrix groups, which are comparatively well-understood, by means of representation theory.[67]

8.9.4 Inﬁnite matrices

It is also possible to consider matrices with infinitely many rows and/or columns[68] even if, being infinite objects, one cannot write down such matrices explicitly. All that matters is that for every element in the set indexing rows, and every element in the set indexing columns, there is a well-defined entry (these index sets need not even be subsets of the natural numbers). The basic operations of addition, subtraction, scalar multiplication and transposition can still be defined without problem; however matrix multiplication may involve infinite summations to define the resulting entries, and these are not defined in general. ⊕ If R is any ring with unity, then the ring of endomorphisms of M = i∈I R as a right R module is isomorphic to the ring of column finite matrices CFMI (R) whose entries are indexed by I × I , and whose columns each contain only finitely many nonzero entries. The endomorphisms of M considered as a left R module result in an analogous object, the row finite matrices RFMI (R) whose rows each only have finitely many nonzero entries. If infinite matrices are used to describe linear maps, then only those matrices can be used all of whose columns have but a finite number of nonzero entries, for the following reason. For a matrix A to describe a linear map f: V→W, bases for both spaces must have been chosen; recall that by definition this means that every vector in the space can be written uniquely as a (finite) linear combination of basis vectors, so that written as a (column) vector v of coefficients, only finitely many entries vi are nonzero. Now the columns of A describe the images by f of individual basis vectors of V in the basis of W, which is only meaningful if these columns have only finitely many nonzero entries. There is no restriction on the rows of A however: in the product A·v there are only finitely many nonzero coefficients of v involved, so every one of its entries, even if it is given as an infinite sum of products, involves only finitely many nonzero terms and is therefore well defined. Moreover this amounts to forming a linear combination of the columns of A that effectively involves only finitely many of them, whence the result has only finitely many nonzero entries, because each of those columns do. One also sees that products of two matrices of the given type is well defined (provided as usual that the column-index and row-index sets match), is again of the same type, and corresponds to the composition of linear maps. If R is a normed ring, then the condition of row or column finiteness can be relaxed. With the norm in place, absolutely convergent series can be used instead of finite sums. For example, the matrices whose column sums are absolutely convergent sequences form a ring. Analogously of course, the matrices whose row sums are absolutely convergent series also form a ring. In that vein, infinite matrices can also be used to describe operators on Hilbert spaces, where convergence and continuity questions arise, which again results in certain constraints that have to be imposed. However, the explicit point of view of matrices tends to obfuscate the matter,[nb 3] and the abstract and more powerful tools of functional analysis can be used instead.

8.9.5 Empty matrices

An empty matrix is a matrix in which the number of rows or columns (or both) is zero.[69][70] Empty matrices help dealing with maps involving the zero vector space. For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them. The determinant of the 0-by-0 matrix is 1 as follows from regarding the empty product occurring in the Leibniz formula for the determinant as 1. This value is also consistent with the fact that the identity map from any ﬁnite dimensional space to itself has determinant 1, a fact that is often used as a part of the characterization of determinants. 78 CHAPTER 8. MATRIX (MATHEMATICS)

8.10 Applications

There are numerous applications of matrices, both in mathematics and other sciences. Some of them merely take advantage of the compact representation of a set of numbers in a matrix. For example, in game theory and economics, the payoff matrix encodes the payoff for two players, depending on which out of a given (finite) set of alternatives the players choose.[71] Text mining and automated thesaurus compilation makes use of document-term matrices such as tf-idf to track frequencies of certain words in several documents.[72] Complex numbers can be represented by particular real 2-by-2 matrices via

[ ] a −b a + ib ↔ , b a

under which addition and multiplication of complex numbers and matrices correspond to each other. For example, 2-by-2 rotation matrices represent the multiplication with some complex number of absolute value 1, as above.A similar interpretation is possible for quaternions[73] and Cliﬀord algebras in general. Early encryption techniques such as the Hill cipher also used matrices. However, due to the linear nature of matrices, these codes are comparatively easy to break.[74] Computer graphics uses matrices both to represent objects and to calculate transformations of objects using aﬃne rotation matrices to accomplish tasks such as projecting a three- dimensional object onto a two-dimensional screen, corresponding to a theoretical camera observation.[75] Matrices over a polynomial ring are important in the study of control theory. Chemistry makes use of matrices in various ways, particularly since the use of quantum theory to discuss molecular bonding and spectroscopy. Examples are the overlap matrix and the Fock matrix used in solving the Roothaan equations to obtain the molecular orbitals of the Hartree–Fock method.

8.10.1 Graph theory

The adjacency matrix of a finite graph is a basic notion of graph theory.[76] It records which vertices of the graph are connected by an edge. Matrices containing just two different values (1 and 0 meaning for example “yes” and “no”, respectively) are called logical matrices. The distance (or cost) matrix contains information about distances of the edges.[77] These concepts can be applied to websites connected by hyperlinks or cities connected by roads etc., in which case (unless the connection network is extremely dense) the matrices tend to be sparse, i.e., contain few nonzero entries. Therefore, specifically tailored matrix algorithms can be used in network theory.

8.10.2 Analysis and geometry

The Hessian matrix of a diﬀerentiable function ƒ: Rn → R consists of the second derivatives of ƒ with respect to the several coordinate directions, i.e.[78]

[ ] ∂2f H(f) = . ∂xi ∂xj

It encodes information about the local growth behaviour of the function: given a critical point x = (x1, ..., xn), i.e., a point where the first partial derivatives ∂f/∂xi of ƒ vanish, the function has a local minimum if the Hessian matrix is positive definite. Quadratic programming can be used to find global minima or maxima of quadratic functions closely related to the ones attached to matrices (see above).[79] Another matrix frequently used in geometrical situations is the Jacobi matrix of a differentiable map f: Rn → Rm. If [80] f1, ..., fm denote the components of f, then the Jacobi matrix is defined as

[ ] ∂f J(f) = i . ∂xj 1≤i≤m,1≤j≤n If n > m, and if the rank of the Jacobi matrix attains its maximal value m, f is locally invertible at that point, by the implicit function theorem.[81] 8.10. APPLICATIONS 79

2 3

  1 1 0 An undirected graph with adjacency matrix 1 0 1. 0 1 0

Partial differential equations can be classified by considering the matrix of coefficients of the highest-order differential operators of the equation. For elliptic partial differential equations this matrix is positive definite, which has decisive influence on the set of possible solutions of the equation in question.[82] The finite element method is an important numerical method to solve partial differential equations, widely applied in simulating complex physical systems. It attempts to approximate the solution to some equation by piecewise linear functions, where the pieces are chosen with respect to a sufficiently fine grid, which in turn can be recast as a matrix equation.[83]

8.10.3 Probability theory and statistics

Stochastic matrices are square matrices whose rows are probability vectors, i.e., whose entries are non-negative and sum up to one. Stochastic matrices are used to define Markov chains with finitely many states.[84] A row of the stochastic matrix gives the probability distribution for the next position of some particle currently in the state that corresponds to the row. Properties of the Markov chain like absorbing states, i.e., states that any particle attains eventually, can be read off the eigenvectors of the transition matrices.[85] Statistics also makes use of matrices in many different forms.[86] Descriptive statistics is concerned with describing data sets, which can often be represented as data matrices, which may then be subjected to dimensionality reduction techniques. The covariance matrix encodes the mutual variance of several random variables.[87] Another technique 80 CHAPTER 8. MATRIX (MATHEMATICS)

[ ] 2 0 At the saddle point (x = 0, y = 0) (red) of the function f(x,−y) = x2 − y2, the Hessian matrix is indeﬁnite. 0 −2

using matrices are linear least squares, a method that approximates a ﬁnite set of pairs (x1, y1), (x2, y2), ..., (xN, yN), by a linear function

yi ≈ axi + b, i = 1, ..., N

which can be formulated in terms of matrices, related to the singular value decomposition of matrices.[88] Random matrices are matrices whose entries are random numbers, subject to suitable probability distributions, such as matrix normal distribution. Beyond probability theory, they are applied in domains ranging from number theory to physics.[89][90]

8.10.4 Symmetries and transformations in physics

Further information: Symmetry in physics

Linear transformations and the associated symmetries play a key role in modern physics. For example, elementary particles in quantum field theory are classified as representations of the Lorentz group of special relativity and, more specifically, by their behavior under the spin group. Concrete representations involving the Pauli matrices and more general gamma matrices are an integral part of the physical description of fermions, which behave as spinors.[91] For the three lightest quarks, there is a group-theoretical representation involving the special unitary group SU(3); for their calculations, physicists use a convenient matrix representation known as the Gell-Mann matrices, which are also used for the SU(3) gauge group that forms the basis of the modern description of strong nuclear interactions, quantum chromodynamics. The Cabibbo–Kobayashi–Maskawa matrix, in turn, expresses the fact that the basic quark states that are important for weak interactions are not the same as, but linearly related to the basic quark states that define particles with specific and distinct masses.[92] 8.10. APPLICATIONS 81

Two diﬀerent Markov chains. The chart depicts the number of[ particles] (of a total[ of 1000)] in state “2”. Both limiting values can be .7 0 .7 .2 determined from the transition matrices, which are given by (red) and (black). .3 1 .3 .8

8.10.5 Linear combinations of quantum states

The ﬁrst model of quantum mechanics (Heisenberg, 1925) represented the theory’s operators by inﬁnite-dimensional matrices acting on quantum states.[93] This is also referred to as matrix mechanics. One particular example is the density matrix that characterizes the “mixed” state of a quantum system as a linear combination of elementary, “pure” eigenstates.[94] Another matrix serves as a key tool for describing the scattering experiments that form the cornerstone of experimental particle physics: Collision reactions such as occur in particle accelerators, where non-interacting particles head towards each other and collide in a small interaction zone, with a new set of non-interacting particles as the result, can be described as the scalar product of outgoing particle states and a linear combination of ingoing particle states. The linear combination is given by a matrix known as the S-matrix, which encodes all information about the possible interactions between particles.[95]

8.10.6 Normal modes

A general application of matrices in physics is to the description of linearly coupled harmonic systems. The equations of motion of such systems can be described in matrix form, with a mass matrix multiplying a generalized velocity to give the kinetic term, and a force matrix multiplying a displacement vector to characterize the interactions. The best way to obtain solutions is to determine the system’s eigenvectors, its normal modes, by diagonalizing the matrix equation. Techniques like this are crucial when it comes to the internal dynamics of molecules: the internal vibrations of systems consisting of mutually bound component atoms.[96] They are also needed for describing mechanical vibrations, and oscillations in electrical circuits.[97] 82 CHAPTER 8. MATRIX (MATHEMATICS)

8.10.7 Geometrical optics

Geometrical optics provides further matrix applications. In this approximative theory, the wave nature of light is neglected. The result is a model in which light rays are indeed geometrical rays. If the deflection of light rays by optical elements is small, the action of a lens or reflective element on a given light ray can be expressed as multiplication of a two-component vector with a two-by-two matrix called ray transfer matrix: the vector’s components are the light ray’s slope and its distance from the optical axis, while the matrix encodes the properties of the optical element. Actually, there are two kinds of matrices, viz. a refraction matrix describing the refraction at a lens surface, and a translation matrix, describing the translation of the plane of reference to the next refracting surface, where another refraction matrix applies. The optical system, consisting of a combination of lenses and/or reflective elements, is simply described by the matrix resulting from the product of the components’ matrices.[98]

8.10.8 Electronics

Traditional mesh analysis in electronics leads to a system of linear equations that can be described with a matrix. The behaviour of many electronic components can be described using matrices. Let A be a 2-dimensional vector with the component’s input voltage v1 and input current i1 as its elements, and let B be a 2-dimensional vector with the component’s output voltage v2 and output current i2 as its elements. Then the behaviour of the electronic component can be described by B = H · A, where H is a 2 x 2 matrix containing one impedance element (h12), one admittance element (h21) and two dimensionless elements (h11 and h22). Calculating a circuit now reduces to multiplying matrices.

8.11 History

Matrices have a long history of application in solving linear equations but they were known as arrays until the 1800s. The Chinese text The Nine Chapters on the Mathematical Art written in 10th–2nd century BCE is the ﬁrst example of the use of array methods to solve simultaneous equations,[99] including the concept of determinants. In 1545 Italian mathematician Girolamo Cardano brought the method to Europe when he published Ars Magna.[100] The Japanese mathematician Seki used the same array methods to solve simultaneous equations in 1683.[101] The Dutch Mathematician Jan de Witt represented transformations using arrays in his 1659 book Elements of Curves (1659).[102] Between 1700 and 1710 Gottfried Wilhelm Leibniz publicized the use of arrays for recording information or solutions and experimented with over 50 diﬀerent systems of arrays.[100] Cramer presented his rule in 1750. The term “matrix” (Latin for “womb”, derived from mater—mother[103]) was coined by James Joseph Sylvester in 1850,[104] who understood a matrix as an object giving rise to a number of determinants today called minors, that is to say, determinants of smaller matrices that derive from the original one by removing columns and rows. In an 1851 paper, Sylvester explains:

I have in previous papers deﬁned a “Matrix” as a rectangular array of terms, out of which diﬀerent systems of determinants may be engendered as from the womb of a common parent.[105]

Arthur Cayley published a treatise on geometric transformations using matrices that were not rotated versions of the coefficients being investigated as had previously been done. Instead he defined operations such as addition, subtraction, multiplication, and division as transformations of those matrices and showed the associative and distributive properties held true. Cayley investigated and demonstrated the non-commutative property of matrix multiplication as well as the commutative property of matrix addition.[100] Early matrix theory had limited the use of arrays almost exclusively to determinants and Arthur Cayley’s abstract matrix operations were revolutionary. He was instrumental in proposing a matrix concept independent of equation systems. In 1858 Cayley published his Memoir on the theory of matrices[106][107] in which he proposed and demonstrated the Cayley-Hamilton theorem.[100] An English mathematician named Cullis was the first to use modern bracket notation for matrices in 1913 and he simultaneously demonstrated the first significant use of the notation A = [ai,j] to represent a matrix where ai,j refers to the ith row and the jth column.[100] The study of determinants sprang from several sources.[108] Number-theoretical problems led Gauss to relate coefficients of quadratic forms, i.e., expressions such as x2 + xy − 2y2, and linear maps in three dimensions to matrices. Eisenstein further developed these notions, including the remark that, in modern parlance, matrix products are 8.12. SEE ALSO 83 non-commutative. Cauchy was the first to prove general statements about determinants, using as definition of the determinant of a matrix A = [ai,j] the following: replace the powers ajk by ajk in the polynomial

∏ a1a2 ··· an (aj − ai) i

8.11.1 Other historical usages of the word “matrix” in mathematics

The word has been used in unusual ways by at least two authors of historical importance. Bertrand Russell and Alfred North Whitehead in their Principia Mathematica (1910–1913) use the word “matrix” in the context of their Axiom of reducibility. They proposed this axiom as a means to reduce any function to one of lower type, successively, so that at the “bottom” (0 order) the function is identical to its extension:

“Let us give the name of matrix to any function, of however many variables, which does not involve any apparent variables. Then any possible function other than a matrix is derived from a matrix by means of generalization, i.e., by considering the proposition which asserts that the function in question is true with all possible values or with some value of one of the arguments, the other argument or arguments remaining undetermined”.[114]

For example a function Φ(x, y) of two variables x and y can be reduced to a collection of functions of a single variable, e.g., y, by “considering” the function for all possible values of “individuals” ai substituted in place of variable x. And then the resulting collection of functions of the single variable y, i.e., ∀aᵢ: Φ(ai, y), can be reduced to a “matrix” of values by “considering” the function for all possible values of “individuals” bi substituted in place of variable y:

∀b∀aᵢ: Φ(ai, b).

Alfred Tarski in his 1946 Introduction to Logic used the word “matrix” synonymously with the notion of truth table as used in mathematical logic.[115]

8.12 See also

• Algebraic multiplicity • Geometric multiplicity • Gram-Schmidt process • List of matrices 84 CHAPTER 8. MATRIX (MATHEMATICS)

• Matrix calculus • Periodic matrix set • Tensor

8.13 Notes

[1] equivalently, table

[2] Anton (1987, p. 23)

[3] Beauregard & Fraleigh (1973, p. 56)

[4] Young, Cynthia. Precalculus. Laurie Rosatone. p. 727. Check date values in: |accessdate= (help);

[5] K. Bryan and T. Leise. The $25,000,000,000 eigenvector: The linear algebra behind Google. SIAM Review, 48(3):569– 581, 2006.

[6] Lang 2002

[7] Fraleigh (1976, p. 209)

[8] Nering (1970, p. 37)

[9] Oualline 2003, Ch. 5

[10] “How to organize, add and multiply matrices - Bill Shillito”. TED ED. Retrieved April 6, 2013.

[11] Brown 1991, Definition I.2.1 (addition), Definition I.2.4 (scalar multiplication), and Definition I.2.33 (transpose)

[12] Brown 1991, Theorem I.2.6

[13] Brown 1991, Deﬁnition I.2.20

[14] Brown 1991, Theorem I.2.24

[15] Horn & Johnson 1985, Ch. 4 and 5

[16] Bronson (1970, p. 16)

[17] Kreyszig (1972, p. 220)

[18] Protter & Morrey (1970, p. 869)

[19] Kreyszig (1972, pp. 241,244)

[20] Schneider, Hans; Barker, George Phillip (2012), Matrices and Linear Algebra, Dover Books on Mathematics, Courier Dover Corporation, p. 251, ISBN 9780486139302.

[21] Perlis, Sam (1991), Theory of Matrices, Dover books on advanced mathematics, Courier Dover Corporation, p. 103, ISBN 9780486668109.

[22] Anton, Howard (414), Elementary Linear Algebra (10th ed.), John Wiley & Sons, ISBN 9780470458211 .

[23] Horn, Roger A.; Johnson, Charles R. (2012), Matrix Analysis (2nd ed.), Cambridge University Press, p. 17, ISBN 9780521839402.

[24] Brown 1991, I.2.21 and 22

[25] Greub 1975, Section III.2

[26] Brown 1991, Deﬁnition II.3.3

[27] Greub 1975, Section III.1

[28] Brown 1991, Theorem II.3.22

[29] Horn & Johnson 1985, Theorem 2.5.6

[30] Brown 1991, Deﬁnition I.2.28 8.13. NOTES 85

[31] Brown 1991, Deﬁnition I.5.13

[32] Horn & Johnson 1985, Chapter 7

[33] Horn & Johnson 1985, Theorem 7.2.1

[34] Horn & Johnson 1985, Example 4.0.6, p. 169

[35] Brown 1991, Deﬁnition III.2.1

[36] Brown 1991, Theorem III.2.12

[37] Brown 1991, Corollary III.2.16

[38] Mirsky 1990, Theorem 1.4.1

[39] Brown 1991, Theorem III.3.18

[40] Brown 1991, Deﬁnition III.4.1

[41] Brown 1991, Deﬁnition III.4.9

[42] Brown 1991, Corollary III.4.10

[43] Householder 1975, Ch. 7

[44] Bau III & Trefethen 1997

[45] Golub & Van Loan 1996, Algorithm 1.3.1

[46] Golub & Van Loan 1996, Chapters 9 and 10, esp. section 10.2

[47] Golub & Van Loan 1996, Chapter 2.3

[48] For example, Mathematica, see Wolfram 2003, Ch. 3.7

[49] Press, Flannery & Teukolsky 1992

[50] Stoer & Bulirsch 2002, Section 4.1

[51] Horn & Johnson 1985, Theorem 2.5.4

[52] Horn & Johnson 1985, Ch. 3.1, 3.2

[53] Arnold & Cooke 1992, Sections 14.5, 7, 8

[54] Bronson 1989, Ch. 15

[55] Coburn 1955, Ch. V

[56] Lang 2002, Chapter XIII

[57] Lang 2002, XVII.1, p. 643

[58] Lang 2002, Proposition XIII.4.16

[59] Reichl 2004, Section L.2

[60] Greub 1975, Section III.3

[61] Greub 1975, Section III.3.13

[62] See any standard reference in group.

[63] Baker 2003, Def. 1.30

[64] Baker 2003, Theorem 1.2

[65] Artin 1991, Chapter 4.5

[66] Rowen 2008, Example 19.2, p. 198

[67] See any reference in representation theory or group representation.

[68] See the item “Matrix” in Itõ, ed. 1987 86 CHAPTER 8. MATRIX (MATHEMATICS)

[69] “Empty Matrix: A matrix is empty if either its row or column dimension is zero”, Glossary, O-Matrix v6 User Guide

[70] “A matrix having at least one dimension equal to zero is called an empty matrix”, MATLAB Data Structures

[71] Fudenberg & Tirole 1983, Section 1.1.1

[72] Manning 1999, Section 15.3.4

[73] Ward 1997, Ch. 2.8

[74] Stinson 2005, Ch. 1.1.5 and 1.2.4

[75] Association for Computing Machinery 1979, Ch. 7

[76] Godsil & Royle 2004, Ch. 8.1

[77] Punnen 2002

[78] Lang 1987a, Ch. XVI.6

[79] Nocedal 2006, Ch. 16

[80] Lang 1987a, Ch. XVI.1

[81] Lang 1987a, Ch. XVI.5. For a more advanced, and more general statement see Lang 1969, Ch. VI.2

[82] Gilbarg & Trudinger 2001

[83] Šolin 2005, Ch. 2.5. See also stiﬀness method.

[84] Latouche & Ramaswami 1999

[85] Mehata & Srinivasan 1978, Ch. 2.8

[86] Healy, Michael (1986), Matrices for Statistics, Oxford University Press, ISBN 978-0-19-850702-4

[87] Krzanowski 1988, Ch. 2.2., p. 60

[88] Krzanowski 1988, Ch. 4.1

[89] Conrey 2007

[90] Zabrodin, Brezin & Kazakov et al. 2006

[91] Itzykson & Zuber 1980, Ch. 2

[92] see Burgess & Moore 2007, section 1.6.3. (SU(3)), section 2.4.3.2. (Kobayashi–Maskawa matrix)

[93] Schiﬀ 1968, Ch. 6

[94] Bohm 2001, sections II.4 and II.8

[95] Weinberg 1995, Ch. 3

[96] Wherrett 1987, part II

[97] Riley, Hobson & Bence 1997, 7.17

[98] Guenther 1990, Ch. 5

[99] Shen, Crossley & Lun 1999 cited by Bretscher 2005, p. 1

[100] Discrete Mathematics 4th Ed. Dossey, Otto, Spense, Vanden Eynden, Published by Addison Wesley, October 10, 2001 ISBN 978-0321079121 | p.564-565

[101] Needham, Joseph; Wang Ling (1959). Science and Civilisation in China III. Cambridge: Cambridge University Press. p. 117. ISBN 9780521058018.

[102] Discrete Mathematics 4th Ed. Dossey, Otto, Spense, Vanden Eynden, Published by Addison Wesley, October 10, 2001 ISBN 978-0321079121 | p.564

[103] Merriam–Webster dictionary, Merriam–Webster, retrieved April 20, 2009 8.14. REFERENCES 87

[104] Although many sources state that J. J. Sylvester coined the mathematical term “matrix” in 1848, Sylvester published nothing in 1848. (For proof that Sylvester published nothing in 1848, see: J. J. Sylvester with H. F. Baker, ed., The Collected Mathematical Papers of James Joseph Sylvester (Cambridge, England: Cambridge University Press, 1904), vol. 1.) His earliest use of the term “matrix” occurs in 1850 in: J. J. Sylvester (1850) “Additions to the articles in the September number of this journal, “On a new class of theorems,” and on Pascal’s theorem,” The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 37 : 363-370. From page 369: “For this purpose we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants … "

[105] The Collected Mathematical Papers of James Joseph Sylvester: 1837–1853, Paper 37, p. 247

[106] Phil.Trans. 1858, vol.148, pp.17-37 Math. Papers II 475-496

[107] Dieudonné, ed. 1978, Vol. 1, Ch. III, p. 96

[108] Knobloch 1994

[109] Hawkins 1975

[110] Kronecker 1897

[111] Weierstrass 1915, pp. 271–286

[112] Bôcher 2004

[113] Mehra & Rechenberg 1987

[114] Whitehead, Alfred North; and Russell, Bertrand (1913) Principia Mathematica to *56, Cambridge at the University Press, Cambridge UK (republished 1962) cf page 162ﬀ.

[115] Tarski, Alfred; (1946) Introduction to Logic and the Methodology of Deductive Sciences, Dover Publications, Inc, New York NY, ISBN 0-486-28462-X.

[1] Eigen means “own” in German and in Dutch.

[2] Additionally, the group is required to be closed in the general linear group.

[3] “Not much of matrix theory carries over to inﬁnite-dimensional spaces, and what does is not so useful, but it sometimes helps.” Halmos 1982, p. 23, Chapter 5

8.14 References

• Anton, Howard (1987), Elementary Linear Algebra (5th ed.), New York: Wiley, ISBN 0-471-84819-0 • Arnold, Vladimir I.; Cooke, Roger (1992), Ordinary diﬀerential equations, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-3-540-54813-3 • Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1 • Association for Computing Machinery (1979), Computer Graphics, Tata McGraw–Hill, ISBN 978-0-07-059376- 3 • Baker, Andrew J. (2003), Matrix Groups: An Introduction to Lie Group Theory, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-1-85233-470-3 • Bau III, David; Trefethen, Lloyd N. (1997), Numerical linear algebra, Philadelphia, PA: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-361-9 • Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduc- tion to Groups, Rings, and Fields, Boston: Houghton Miﬄin Co., ISBN 0-395-14017-X • Bretscher, Otto (2005), Linear Algebra with Applications (3rd ed.), Prentice Hall • Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490 • Bronson, Richard (1989), Schaum’s outline of theory and problems of matrix operations, New York: McGraw– Hill, ISBN 978-0-07-007978-6 88 CHAPTER 8. MATRIX (MATHEMATICS)

• Brown, William C. (1991), Matrices and vector spaces, New York, NY: Marcel Dekker, ISBN 978-0-8247- 8419-5

• Coburn, Nathaniel (1955), Vector and tensor analysis, New York, NY: Macmillan, OCLC 1029828

• Conrey, J. Brian (2007), Ranks of elliptic curves and random matrix theory, Cambridge University Press, ISBN 978-0-521-69964-8

• Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: Addison-Wesley, ISBN 0- 201-01984-1

• Fudenberg, Drew; Tirole, Jean (1983), Game Theory, MIT Press

• Gilbarg, David; Trudinger, Neil S. (2001), Elliptic partial diﬀerential equations of second order (2nd ed.), Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-3-540-41160-4

• Godsil, Chris; Royle, Gordon (2004), Algebraic Graph Theory, Graduate Texts in Mathematics 207, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-95220-8

• Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Johns Hopkins, ISBN 978-0- 8018-5414-9

• Greub, Werner Hildbert (1975), Linear algebra, Graduate Texts in Mathematics, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-90110-7

• Halmos, Paul Richard (1982), A Hilbert space problem book, Graduate Texts in Mathematics 19 (2nd ed.), Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-90685-0, MR 675952

• Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521- 38632-6

• Householder, Alston S. (1975), The theory of matrices in numerical analysis, New York, NY: Dover Publica- tions, MR 0378371

• Kreyszig, Erwin (1972), Advanced Engineering Mathematics (3rd ed.), New York: Wiley, ISBN 0-471-50728- 8.

• Krzanowski, Wojtek J. (1988), Principles of multivariate analysis, Oxford Statistical Science Series 3, The Clarendon Press Oxford University Press, ISBN 978-0-19-852211-9, MR 969370

• Itõ, Kiyosi, ed. (1987), Encyclopedic dictionary of mathematics. Vol. I-IV (2nd ed.), MIT Press, ISBN 978-0- 262-09026-1, MR 901762

• Lang, Serge (1969), Analysis II, Addison-Wesley

• Lang, Serge (1987a), Calculus of several variables (3rd ed.), Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-96405-8

• Lang, Serge (1987b), Linear algebra, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-96412-6

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer- Verlag, ISBN 978-0-387-95385-4, MR 1878556

• Latouche, Guy; Ramaswami, Vaidyanathan (1999), Introduction to matrix analytic methods in stochastic modeling (1st ed.), Philadelphia, PA: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-425-8

• Manning, Christopher D.; Schütze, Hinrich (1999), Foundations of statistical natural language processing, MIT Press, ISBN 978-0-262-13360-9

• Mehata, K. M.; Srinivasan, S. K. (1978), Stochastic processes, New York, NY: McGraw–Hill, ISBN 978-0-07- 096612-3

• Mirsky, Leonid (1990), An Introduction to Linear Algebra, Courier Dover Publications, ISBN 978-0-486- 66434-7

• Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76-91646 8.14. REFERENCES 89

• Nocedal, Jorge; Wright, Stephen J. (2006), Numerical Optimization (2nd ed.), Berlin, DE; New York, NY: Springer-Verlag, p. 449, ISBN 978-0-387-30303-1

• Oualline, Steve (2003), Practical C++ programming, O'Reilly, ISBN 978-0-596-00419-4

• Press, William H.; Flannery, Brian P.; Teukolsky, Saul A.; Vetterling, William T. (1992), “LU Decomposi- tion and Its Applications”, Numerical Recipes in FORTRAN: The Art of Scientiﬁc Computing (PDF) (2nd ed.), Cambridge University Press, pp. 34–42

• Protter, Murray H.; Morrey, Jr., Charles B. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, LCCN 76087042

• Punnen, Abraham P.; Gutin, Gregory (2002), The traveling salesman problem and its variations, Boston, MA: Kluwer Academic Publishers, ISBN 978-1-4020-0664-7

• Reichl, Linda E. (2004), The transition to chaos: conservative classical systems and quantum manifestations, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-98788-0

• Rowen, Louis Halle (2008), Graduate Algebra: noncommutative view, Providence, RI: American Mathematical Society, ISBN 978-0-8218-4153-2

• Šolin, Pavel (2005), Partial Diﬀerential Equations and the Finite Element Method, Wiley-Interscience, ISBN 978-0-471-76409-0

• Stinson, Douglas R. (2005), Cryptography, Discrete Mathematics and its Applications, Chapman & Hall/CRC, ISBN 978-1-58488-508-5

• Stoer, Josef; Bulirsch, Roland (2002), Introduction to Numerical Analysis (3rd ed.), Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-95452-3

• Ward, J. P. (1997), Quaternions and Cayley numbers, Mathematics and its Applications 403, Dordrecht, NL: Kluwer Academic Publishers Group, ISBN 978-0-7923-4513-8, MR 1458894

• Wolfram, Stephen (2003), The Mathematica Book (5th ed.), Champaign, IL: Wolfram Media, ISBN 978-1- 57955-022-6

8.14.1 Physics references

• Bohm, Arno (2001), Quantum Mechanics: Foundations and Applications, Springer, ISBN 0-387-95330-2

• Burgess, Cliﬀ; Moore, Guy (2007), The Standard Model. A Primer, Cambridge University Press, ISBN 0-521- 86036-9

• Guenther, Robert D. (1990), Modern Optics, John Wiley, ISBN 0-471-60538-7

• Itzykson, Claude; Zuber, Jean-Bernard (1980), Quantum Field Theory, McGraw–Hill, ISBN 0-07-032071-3

• Riley, Kenneth F.; Hobson, Michael P.; Bence, Stephen J. (1997), Mathematical methods for physics and engineering, Cambridge University Press, ISBN 0-521-55506-X

• Schiﬀ, Leonard I. (1968), Quantum Mechanics (3rd ed.), McGraw–Hill

• Weinberg, Steven (1995), The Quantum Theory of Fields. Volume I: Foundations, Cambridge University Press, ISBN 0-521-55001-7

• Wherrett, Brian S. (1987), Group Theory for Atoms, Molecules and Solids, Prentice–Hall International, ISBN 0-13-365461-3

• Zabrodin, Anton; Brezin, Édouard; Kazakov, Vladimir; Serban, Didina; Wiegmann, Paul (2006), Applications of Random Matrices in Physics (NATO Science Series II: Mathematics, Physics and Chemistry), Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-1-4020-4530-1 90 CHAPTER 8. MATRIX (MATHEMATICS)

8.14.2 Historical references

• A. Cayley A memoir on the theory of matrices. Phil. Trans. 148 1858 17-37; Math. Papers II 475-496 • Bôcher, Maxime (2004), Introduction to higher algebra, New York, NY: Dover Publications, ISBN 978-0-486- 49570-5, reprint of the 1907 original edition • Cayley, Arthur (1889), The collected mathematical papers of Arthur Cayley, I (1841–1853), Cambridge Uni- versity Press, pp. 123–126 • Dieudonné, Jean, ed. (1978), Abrégé d'histoire des mathématiques 1700-1900, Paris, FR: Hermann • Hawkins, Thomas (1975), “Cauchy and the spectral theory of matrices”, Historia Mathematica 2: 1–29, doi:10.1016/0315-0860(75)90032-4, ISSN 0315-0860, MR 0469635 • Knobloch, Eberhard (1994), “From Gauss to Weierstrass: determinant theory and its historical evaluations”, The intersection of history and mathematics, Science Networks Historical Studies 15, Basel, Boston, Berlin: Birkhäuser, pp. 51–66, MR 1308079 • Kronecker, Leopold (1897), Hensel, Kurt, ed., Leopold Kronecker’s Werke, Teubner • Mehra, Jagdish; Rechenberg, Helmut (1987), The Historical Development of Quantum Theory (1st ed.), Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-96284-9 • Shen, Kangshen; Crossley, John N.; Lun, Anthony Wah-Cheung (1999), Nine Chapters of the Mathematical Art, Companion and Commentary (2nd ed.), Oxford University Press, ISBN 978-0-19-853936-0 • Weierstrass, Karl (1915), Collected works 3

8.15 External links

Encyclopedic articles

• Hazewinkel, Michiel, ed. (2001), “Matrix”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

History

• MacTutor: Matrices and determinants • Matrices and Linear Algebra on the Earliest Uses Pages • Earliest Uses of Symbols for Matrices and Vectors

Online books

• Kaw, Autar K., Introduction to Matrix Algebra, ISBN 978-0-615-25126-4 • The Matrix Cookbook (PDF), retrieved 24 March 2014 • Brookes, Mike (2005), The Matrix Reference Manual, London: Imperial College, retrieved 10 Dec 2008

Online matrix calculators

• SimplyMath (Matrix Calculator) • Matrix Calculator (DotNumerics) • Xiao, Gang, Matrix calculator, retrieved 10 Dec 2008 • Online matrix calculator, retrieved 10 Dec 2008 • Online matrix calculator (ZK framework), retrieved 26 Nov 2009 8.15. EXTERNAL LINKS 91

• Oehlert, Gary W.; Bingham, Christopher, MacAnova, University of Minnesota, School of Statistics, retrieved 10 Dec 2008, a freeware package for matrix algebra and statistics • Online matrix calculator, retrieved 14 Dec 2009

• Operation with matrices in R (determinant, track, inverse, adjoint, transpose) Chapter 9

Numerical linear algebra

Numerical linear algebra is the study of algorithms for performing linear algebra computations, most notably matrix operations, on computers. It is often a fundamental part of engineering and computational science problems, such as image and signal processing, telecommunication, computational finance, materials science simulations, structural biology, data mining, bioinformatics, fluid dynamics, and many other areas. Such software relies heavily on the development, analysis, and implementation of state-of-the-art algorithms for solving various numerical linear algebra problems, in large part because of the role of matrices in finite difference and finite element methods. Common problems in numerical linear algebra include computing the following: LU decomposition, QR decomposition, singular value decomposition, eigenvalues.

9.1 See also

• Iterative methods

• Numerical analysis, of which numerical linear algebra is a subspecialty

• Gaussian elimination, an important algorithm in numerical linear algebra

• BLAS and LAPACK, highly optimized computer libraries which implement most basic algorithms in numerical linear algebra

• List of numerical analysis software

• List of numerical libraries

9.2 References

• Leader, Jeﬀery J. (2004). Numerical Analysis and Scientiﬁc Computation. Addison Wesley. ISBN 0-201- 73499-0.

• Bau III, David; Trefethen, Lloyd N. (1997). Numerical linear algebra. Philadelphia: Society for Industrial and Applied Mathematics. ISBN 978-0-89871-361-9.

• J. H. Wilkinson and C. Reinsch, “Linear Algebra, volume II of Handbook for Automatic Computation” SIAM Review 14, 658 (1972).

• Golub, Gene H.; van Loan, Charles F. (1996), Matrix Computations, 3rd edition, Johns Hopkins University Press, ISBN 978-0-8018-5414-9

92 9.3. EXTERNAL LINKS 93

9.3 External links

• Freely available software for numerical algebra on the web, composed by Jack Dongarra and Hatem Ltaief, University of Tennessee • NAG Library of numerical linear algebra routines Chapter 10

System of linear equations

A linear system in three variables determines a collection of planes. The intersection point is the solution.

In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables.[1] For example,

94 10.1. ELEMENTARY EXAMPLE 95

3x + 2y − z = 1 2x − 2y + 4z = −2 − 1 − x + 2 y z = 0 is a system of three equations in the three variables x, y, z.A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisﬁed. A solution to the system above is given by

x = 1 y = −2 z = −2 since it makes all three equations valid. The word "system" indicates that the equations are to be considered collec- tively, rather than individually. In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics.A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system. Very often, the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers, but the theory and the algorithms apply for coefficients and solutions in any field. For solutions in an integral domain like the ring of the integers, or in other algebraic structures, other theories have been developed, see Linear equation over a ring. Integer linear programming is a collection of methods for finding the “best” integer solution (when there are many). Gröbner basis theory provides algorithms when coefficients and unknowns are polynomials. Also tropical geometry is an example of linear algebra in a more exotic structure.

10.1 Elementary example

The simplest kind of linear system involves two equations and two variables:

2x + 3y = 6 4x + 9y = 15.

One method for solving such a system is as follows. First, solve the top equation for x in terms of y :

3 x = 3 − y. 2 Now substitute this expression for x into the bottom equation:

( ) 3 4 3 − y + 9y = 15. 2

This results in a single equation involving only the variable y . Solving gives y = 1 , and substituting this back into the equation for x yields x = 3/2 . This method generalizes to systems with additional variables (see “elimination of variables” below, or the article on elementary algebra.)

10.2 General form

A general system of m linear equations with n unknowns can be written as 96 CHAPTER 10. SYSTEM OF LINEAR EQUATIONS

a11x1 + a12x2 + ··· + a1nxn = b1

a21x1 + a22x2 + ··· + a2nxn = b2 ......

am1x1 + am2x2 + ··· + amnxn = bm.

Here x1, x2, . . . , xn are the unknowns, a11, a12, . . . , amn are the coeﬃcients of the system, and b1, b2, . . . , bm are the constant terms. Often the coeﬃcients and unknowns are real or complex numbers, but integers and rational numbers are also seen, as are polynomials and elements of an abstract algebraic structure.

10.2.1 Vector equation

One extremely helpful view is that each unknown is a weight for a column vector in a linear combination.

        a11 a12 a1n b1          a21   a22   a2n   b2  x1 .  + x2 .  + ··· + xn .  =  .   .   .   .   .  am1 am2 amn bm

This allows all the language and theory of vector spaces (or more generally, modules) to be brought to bear. For example, the collection of all possible linear combinations of the vectors on the left-hand side is called their span, and the equations have a solution just when the right-hand vector is within that span. If every vector within that span has exactly one expression as a linear combination of the given left-hand vectors, then any solution is unique. In any event, the span has a basis of linearly independent vectors that do guarantee exactly one expression; and the number of vectors in that basis (its dimension) cannot be larger than m or n, but it can be smaller. This is important because if we have m independent vectors a solution is guaranteed regardless of the right-hand side, and otherwise not guaranteed.

10.2.2 Matrix equation

The vector equation is equivalent to a matrix equation of the form

Ax = b

where A is an m×n matrix, x is a column vector with n entries, and b is a column vector with m entries.

      a11 a12 ··· a1n x1 b1        a21 a22 ··· a2n  x2   b2  A =  . . . . , x =  . , b =  .   . . .. .   .   .  am1 am2 ··· amn xn bm

The number of vectors in a basis for the span is now expressed as the rank of the matrix.

10.3 Solution set

A solution of a linear system is an assignment of values to the variables x1, x2, ..., xn such that each of the equations is satisﬁed. The set of all possible solutions is called the solution set. A linear system may behave in any one of three possible ways: 10.3. SOLUTION SET 97

(2,3)

x-y=-1 3x+y=9

The solution set for the equations x − y = −1 and 3x + y = 9 is the single point (2, 3).

1. The system has inﬁnitely many solutions.

2. The system has a single unique solution.

3. The system has no solution.

10.3.1 Geometric interpretation

For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set. For three variables, each linear equation determines a plane in three-dimensional space, and the solution set is the intersection of these planes. Thus the solution set may be a plane, a line, a single point, or the empty set. For example, as three parallel planes do not have a common point, the solution set of their equations is empty; the solution set of the equations of three planes intersecting at a point is single point; if three planes pass through two points, their equations have at least two common solutions; in fact the solution set is inﬁnite and consists in all the line passing through these points.[2] 98 CHAPTER 10. SYSTEM OF LINEAR EQUATIONS

For n variables, each linear equation determines a hyperplane in n-dimensional space. The solution set is the intersection of these hyperplanes, which may be a ﬂat of any dimension.

10.3.2 General behavior

The solution set for two equations in three variables is usually a line.

In general, the behavior of a linear system is determined by the relationship between the number of equations and the number of unknowns:

• Usually, a system with fewer equations than unknowns has inﬁnitely many solutions, but it may have no solution. Such a system is known as an underdetermined system. • Usually, a system with the same number of equations and unknowns has a single unique solution. • Usually, a system with more equations than unknowns has no solution. Such a system is also known as an overdetermined system.

In the ﬁrst case, the dimension of the solution set is usually equal to n − m, where n is the number of variables and m is the number of equations. The following pictures illustrate this trichotomy in the case of two variables: 10.4. PROPERTIES 99

The first system has infinitely many solutions, namely all of the points on the blue line. The second system has a single unique solution, namely the intersection of the two lines. The third system has no solutions, since the three lines share no common point. Keep in mind that the pictures above show only the most common case. It is possible for a system of two equations and two unknowns to have no solution (if the two lines are parallel), or for a system of three equations and two unknowns to be solvable (if the three lines intersect at a single point). In general, a system of linear equations may behave differently from expected if the equations are linearly dependent, or if two or more of the equations are inconsistent.

10.4 Properties

10.4.1 Independence

The equations of a linear system are independent if none of the equations can be derived algebraically from the others. When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set. For linear equations, logical independence is the same as linear independence. For example, the equations

3x + 2y = 6 and 6x + 4y = 12 are not independent — they are the same equation when scaled by a factor of two, and they would produce identical graphs. This is an example of equivalence in a system of linear equations. For a more complicated example, the equations

x − 2y = −1 3x + 5y = 8 4x + 3y = 7 are not independent, because the third equation is the sum of the other two. Indeed, any one of these equations can be derived from the other two, and any one of the equations can be removed without aﬀecting the solution set. The graphs of these equations are three lines that intersect at a single point.

10.4.2 Consistency

See also: Consistent and inconsistent equations A linear system is inconsistent if it has no solution, and otherwise it is said to be consistent . When the system is inconsistent, it is possible to derive a contradiction from the equations, that may always be rewritten such as the statement 0 = 1. For example, the equations

3x + 2y = 6 and 3x + 2y = 12 are inconsistent. In fact, by subtracting the ﬁrst equation from the second one and multiplying both sides of the result by 1/6, we get 0 = 1. The graphs of these equations on the xy-plane are a pair of parallel lines. It is possible for three linear equations to be inconsistent, even though any two of them are consistent together. For example, the equations 100 CHAPTER 10. SYSTEM OF LINEAR EQUATIONS

The equations x − 2y = −1, 3x + 5y = 8, and 4x + 3y = 7 are linearly dependent.

x + y = 1 2x + y = 1 3x + 2y = 3 are inconsistent. Adding the first two equations together gives 3x + 2y = 2, which can be subtracted from the third equation to yield 0 = 1. Note that any two of these equations have a common solution. The same phenomenon can occur for any number of equations. In general, inconsistencies occur if the left-hand sides of the equations in a system are linearly dependent, and the constant terms do not satisfy the dependence relation. A system of equations whose left-hand sides are linearly independent is always consistent. Putting it another way, according to the Rouché–Capelli theorem, any system of equations (overdetermined or otherwise) is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has k free parameters where k is the difference between the number of variables and the rank; hence in such a case there are an infinitude of solutions. The rank of a system of equations can never be higher than [the number of variables] + 1, which means 10.5. SOLVING A LINEAR SYSTEM 101

The equations 3x + 2y = 6 and 3x + 2y = 12 are inconsistent. that a system with any number of equations can always be reduced to a system that has a number of independent equations that is at most equal to [the number of variables] + 1.

10.4.3 Equivalence

Two linear systems using the same set of variables are equivalent if each of the equations in the second system can be derived algebraically from the equations in the ﬁrst system, and vice versa. Two systems are equivalent if either both are inconsistent or each equation of any of them is a linear combination of the equations of the other one. It follows that two linear systems are equivalent if and only if they have the same solution set.

10.5 Solving a linear system

There are several algorithms for solving a system of linear equations. 102 CHAPTER 10. SYSTEM OF LINEAR EQUATIONS

10.5.1 Describing the solution

When the solution set is finite, it is reduced to a single element. In this case, the unique solution is described by a sequence of equations whose left-hand sides are the names of the unknowns and right-hand sides are the corresponding values, for example (x = 3, y = −2, z = 6) . When an order on the unknowns has been fixed, for example the alphabetical order the solution may be described as a vector of values, like (3, −2, 6) for the previous example. It can be difficult to describe a set with infinite solutions. Typically, some of the variables are designated as free (or independent, or as parameters), meaning that they are allowed to take any value, while the remaining variables are dependent on the values of the free variables. For example, consider the following system:

x + 3y − 2z = 5 3x + 5y + 6z = 7

The solution set to this system can be described by the following equations: x = −7z − 1 and y = 3z + 2.

Here z is the free variable, while x and y are dependent on z. Any point in the solution set can be obtained by first choosing a value for z, and then computing the corresponding values for x and y. Each free variable gives the solution space one degree of freedom, the number of which is equal to the dimension of the solution set. For example, the solution set for the above equation is a line, since a point in the solution set can be chosen by specifying the value of the parameter z. An infinite solution of higher order may describe a plane, or higher-dimensional set. Different choices for the free variables may lead to different descriptions of the same solution set. For example, the solution to the above equations can alternatively be described as follows:

3 11 1 1 y = − x + and z = − x − . 7 7 7 7 Here x is the free variable, and y and z are dependent.

10.5.2 Elimination of variables

The simplest method for solving a system of linear equations is to repeatedly eliminate variables. This method can be described as follows:

1. In the ﬁrst equation, solve for one of the variables in terms of the others. 2. Substitute this expression into the remaining equations. This yields a system of equations with one fewer equation and one fewer unknown. 3. Continue until you have reduced the system to a single linear equation. 4. Solve this equation, and then back-substitute until the entire solution is found.

For example, consider the following system:

x + 3y − 2z = 5 3x + 5y + 6z = 7 2x + 4y + 3z = 8

Solving the ﬁrst equation for x gives x = 5 + 2z − 3y, and plugging this into the second and third equation yields 10.5. SOLVING A LINEAR SYSTEM 103

−4y + 12z = −8 −2y + 7z = −2

Solving the ﬁrst of these equations for y yields y = 2 + 3z, and plugging this into the second equation yields z = 2. We now have:

x = 5 + 2z − 3y y = 2 + 3z z = 2

Substituting z = 2 into the second equation gives y = 8, and substituting z = 2 and y = 8 into the ﬁrst equation yields x = −15. Therefore, the solution set is the single point (x, y, z) = (−15, 8, 2).

10.5.3 Row reduction

Main article: Gaussian elimination

In row reduction, the linear system is represented as an augmented matrix:

  1 3 −2 5  3 5 6 7  . 2 4 3 8

This matrix is then modiﬁed using elementary row operations until it reaches reduced row echelon form. There are three types of elementary row operations:

Type 1: Swap the positions of two rows. Type 2: Multiply a row by a nonzero scalar. Type 3: Add to one row a scalar multiple of another.

Because these operations are reversible, the augmented matrix produced always represents a linear system that is equivalent to the original. There are several speciﬁc algorithms to row-reduce an augmented matrix, the simplest of which are Gaussian elimination and Gauss-Jordan elimination. The following computation shows Gauss-Jordan elimination applied to the matrix above:

        1 3 −2 5 1 3 −2 5 1 3 −2 5 1 3 −2 5  3 5 6 7  ∼  0 −4 12 −8  ∼  0 −4 12 −8  ∼  0 1 −3 2  2 4 3 8 2 4 3 8 0 −2 7 −2 0 −2 7 −2         1 3 −2 5 1 3 −2 5 1 3 0 9 1 0 0 −15 ∼  0 1 −3 2  ∼  0 1 0 8  ∼  0 1 0 8  ∼  0 1 0 8  . 0 0 1 2 0 0 1 2 0 0 1 2 0 0 1 2

The last matrix is in reduced row echelon form, and represents the system x = −15, y = 8, z = 2. A comparison with the example in the previous section on the algebraic elimination of variables shows that these two methods are in fact the same; the diﬀerence lies in how the computations are written down.

10.5.4 Cramer’s rule

Main article: Cramer’s rule 104 CHAPTER 10. SYSTEM OF LINEAR EQUATIONS

Cramer’s rule is an explicit formula for the solution of a system of linear equations, with each variable given by a quotient of two determinants. For example, the solution to the system

x + 3y − 2z = 5 3x + 5y + 6z = 7 2x + 4y + 3z = 8

is given by

5 3 −2 1 5 −2 1 3 5

7 5 6 3 7 6 3 5 7

8 4 3 2 8 3 2 4 8 x = , y = , z = . 1 3 −2 1 3 −2 1 3 −2

3 5 6 3 5 6 3 5 6 2 4 3 2 4 3 2 4 3

For each variable, the denominator is the determinant of the matrix of coeﬃcients, while the numerator is the determinant of a matrix in which one column has been replaced by the vector of constant terms. Though Cramer’s rule is important theoretically, it has little practical value for large matrices, since the computation of large determinants is somewhat cumbersome. (Indeed, large determinants are most easily computed using row reduction.) Further, Cramer’s rule has very poor numerical properties, making it unsuitable for solving even small systems reliably, unless the operations are performed in rational arithmetic with unbounded precision.

10.5.5 Matrix solution

If the equation system is expressed in the matrix form Ax = b , the entire solution set can also be expressed in matrix form. If the matrix A is square (has m rows and n=m columns) and has full rank (all m rows are independent), then the system has a unique solution given by x = A−1b where A−1 is the inverse of A. More generally, regardless of whether m=n or not and regardless of the rank of A, all solutions (if any exist) are given using the Moore-Penrose pseudoinverse of A, denoted Ag , as follows:

x = Agb + (I − AgA)w

where w is a vector of free parameters that ranges over all possible n×1 vectors. A necessary and sufficient condition for any solution(s) to exist is that the potential solution obtained using w = 0 satisfy Ax = b — that is, that AAgb = b. If this condition does not hold, the equation system is inconsistent and has no solution. If the condition holds, the system is consistent and at least one solution exists. For example, in the above-mentioned case in which A is square and of full rank, Ag simply equals A−1 and the general solution equation simplifies to x = A−1b + (I − A−1A)w = A−1b + (I − I)w = A−1b as previously stated, where w has completely dropped out of the solution, leaving only a single solution. In other cases, though, w remains and hence an infinitude of potential values of the free parameter vector w give an infinitude of solutions of the equation.

10.5.6 Other methods

While systems of three or four equations can be readily solved by hand (see Cracovian), computers are often used for larger systems. The standard algorithm for solving a system of linear equations is based on Gaussian elimination with some modiﬁcations. Firstly, it is essential to avoid division by small numbers, which may lead to inaccurate results. This can be done by reordering the equations if necessary, a process known as pivoting. Secondly, the algorithm does not exactly do Gaussian elimination, but it computes the LU decomposition of the matrix A. This is mostly an 10.6. HOMOGENEOUS SYSTEMS 105

organizational tool, but it is much quicker if one has to solve several systems with the same matrix A but different vectors b. If the matrix A has some special structure, this can be exploited to obtain faster or more accurate algorithms. For instance, systems with a symmetric positive definite matrix can be solved twice as fast with the Cholesky decomposition. Levinson recursion is a fast method for Toeplitz matrices. Special methods exist also for matrices with many zero elements (so-called sparse matrices), which appear often in applications. A completely different approach is often taken for very large systems, which would otherwise take too much time or memory. The idea is to start with an initial approximation to the solution (which does not have to be accurate at all), and to change this approximation in several steps to bring it closer to the true solution. Once the approximation is sufficiently accurate, this is taken to be the solution to the system. This leads to the class of iterative methods.

10.6 Homogeneous systems

See also: Homogeneous diﬀerential equation

A system of linear equations is homogeneous if all of the constant terms are zero:

a11x1 + a12x2 + ··· + a1nxn = 0

a21x1 + a22x2 + ··· + a2nxn = 0 ......

am1x1 + am2x2 + ··· + amnxn = 0. A homogeneous system is equivalent to a matrix equation of the form

Ax = 0 where A is an m × n matrix, x is a column vector with n entries, and 0 is the zero vector with m entries.

10.6.1 Solution set

Every homogeneous system has at least one solution, known as the zero solution (or trivial solution), which is obtained by assigning the value of zero to each of the variables. If the system has a non-singular matrix (det(A) ≠ 0) then it is also the only solution. If the system has a singular matrix then there is a solution set with an inﬁnite number of solutions. This solution set has the following additional properties:

1. If u and v are two vectors representing solutions to a homogeneous system, then the vector sum u + v is also a solution to the system. 2. If u is a vector representing a solution to a homogeneous system, and r is any scalar, then ru is also a solution to the system.

These are exactly the properties required for the solution set to be a linear subspace of Rn. In particular, the solution set to a homogeneous system is the same as the null space of the corresponding matrix A. A numerical solutions to a homogeneous system can be found with a SVD decomposition.

10.6.2 Relation to nonhomogeneous systems

There is a close relationship between the solutions to a linear system and the solutions to the corresponding homogeneous system:

Ax = b and Ax = 0. 106 CHAPTER 10. SYSTEM OF LINEAR EQUATIONS

Speciﬁcally, if p is any speciﬁc solution to the linear system Ax = b, then the entire solution set can be described as

{p + v : v to solution any is Ax = 0} . Geometrically, this says that the solution set for Ax = b is a translation of the solution set for Ax = 0. Specifically, the flat for the first system can be obtained by translating the linear subspace for the homogeneous system by the vector p. This reasoning only applies if the system Ax = b has at least one solution. This occurs if and only if the vector b lies in the image of the linear transformation A.

10.7 See also

• Linear equation over a ring • Arrangement of hyperplanes • Iterative reﬁnement • LAPACK (the free standard package to solve linear equations numerically; available in Fortran, C, C++) • Linear least squares • Matrix decomposition • Matrix splitting • NAG Numerical Library (NAG Library versions of LAPACK solvers) • Row reduction • Simultaneous equations

10.8 Notes

[1] The subject of this article is basic in mathematics, and is treated in a lot of textbooks. Among them, Lay 2005, Meyer 2001, and Strang 2005 contain the material of this article. [2] Charles G. Cullen (1990). Matrices and Linear Transformations. MA: Dover. p. 3. ISBN 978-0-486-66328-9.

10.9 References

See also: Linear algebra § Further reading

10.9.1 Textbooks

• Axler, Sheldon Jay (1997), Linear Algebra Done Right (2nd ed.), Springer-Verlag, ISBN 0-387-98259-0 • Lay, David C. (August 22, 2005), Linear Algebra and Its Applications (3rd ed.), Addison Wesley, ISBN 978- 0-321-28713-7 • Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), ISBN 978-0-89871-454-8 • Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Brooks/Cole, ISBN 0-534-99845-3 • Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International • Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall • Strang, Gilbert (2005), Linear Algebra and Its Applications Chapter 11

Vector space

This article is about linear (vector) spaces. For the structure in incidence geometry, see Linear space (geometry). A vector space (also called a linear space) is a collection of objects called vectors, which may be added together

v v+w w

v v+2w 2w

Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w. and multiplied (“scaled”) by numbers, called scalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below. Euclidean vectors are an example of a vector space. They represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces. Vectors in vector spaces do not necessarily have to be arrow-like objects as they appear in the mentioned examples: vectors are regarded as abstract mathematical objects with particular properties, which in some cases can be visualized as arrows. Vector spaces are the subject of linear algebra and are well understood from this point of view since vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. A vector space may be endowed with additional structure, such as a norm or inner product. Such spaces arise naturally in mathematical analysis, mainly in the guise of infinite-dimensional function spaces whose vectors are functions. Analytical problems call for the ability to decide whether a sequence of vectors converges to a given vector. This is accomplished by considering vector spaces with additional structure, mostly spaces endowed with a

107 108 CHAPTER 11. VECTOR SPACE

suitable topology, thus allowing the consideration of proximity and continuity issues. These topological vector spaces, in particular Banach spaces and Hilbert spaces, have a richer theory. Historically, the first ideas leading to vector spaces can be traced back as far as 17th century’s analytic geometry, matrices, systems of linear equations, and Euclidean vectors. The modern, more abstract treatment, first formulated by Giuseppe Peano in 1888, encompasses more general objects than Euclidean space, but much of the theory can be seen as an extension of classical geometric ideas like lines, planes and their higher-dimensional analogs. Today, vector spaces are applied throughout mathematics, science and engineering. They are the appropriate linear- algebraic notion to deal with systems of linear equations; offer a framework for Fourier expansion, which is employed in image compression routines; or provide an environment that can be used for solution techniques for partial differential equations. Furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra.

11.1 Introduction and deﬁnition

The concept of vector space will ﬁrst be explained by describing two particular examples:

11.1.1 First example: arrows in the plane

The first example of a vector space consists of arrows in a fixed plane, starting at one fixed point. This is used in physics to describe forces or velocities. Given any two such arrows, v and w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows and is denoted v + w. In the special case of two arrows on the same line, their sum is the arrow on this line whose length is the sum or the difference of the lengths, depending on whether the arrows have the same direction. Another operation that can be done with arrows is scaling: given any positive real number a, the arrow that has the same direction as v, but is dilated or shrunk by multiplying its length by a, is called multiplication of v by a. It is denoted av. When a is negative, av is defined as the arrow pointing in the opposite direction, instead. The following shows a few examples: if a = 2, the resulting vector aw has the same direction as w, but is stretched to the double length of w (right image below). Equivalently 2w is the sum w + w. Moreover, (−1)v = −v has the opposite direction and the same length as v (blue vector pointing down in the right image).

11.1.2 Second example: ordered pairs of numbers

A second key example of a vector space is provided by pairs of real numbers x and y. (The order of the components x and y is signiﬁcant, so such a pair is also called an ordered pair.) Such a pair is written as (x, y). The sum of two such pairs and multiplication of a pair with a number is deﬁned as follows:

(x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)

and

a (x, y) = (ax, ay).

The ﬁrst example above reduces to this one if the arrows are represented by the pair of Cartesian coordinates of their end points.

11.1.3 Deﬁnition

A vector space over a ﬁeld F is a set V together with two operations that satisfy the eight axioms listed below. Elements of V are commonly called vectors. Elements of F are commonly called scalars. The ﬁrst operation, called vector addition or simply addition, takes any two vectors v and w and assigns to them a third vector which is commonly 11.2. HISTORY 109

written as v + w, and called the sum of these two vectors. The second operation, called scalar multiplication takes any scalar a and any vector v and gives another vector av. In this article, vectors are distinguished from scalars by boldface.[nb 1] In the two examples above, the field is the field of the real numbers and the set of the vectors consists of the planar arrows with fixed starting point and of pairs of real numbers, respectively. To qualify as a vector space, the set V and the operations of addition and multiplication must adhere to a number of requirements called axioms.[1] In the list below, let u, v and w be arbitrary vectors in V, and a and b scalars in F. These axioms generalize properties of the vectors introduced in the above examples. Indeed, the result of addition of two ordered pairs (as in the second example above) does not depend on the order of the summands:

(xᵥ, yᵥ) + (x, y) = (x, y) + (xᵥ, yᵥ).

Likewise, in the geometric example of vectors as arrows, v + w = w + v since the parallelogram defining the sum of the vectors is independent of the order of the vectors. All other axioms can be checked in a similar manner in both examples. Thus, by disregarding the concrete nature of the particular type of vectors, the definition incorporates these two and many more examples in one notion of vector space. Subtraction of two vectors and division by a (non-zero) scalar can be defined as

v − w = v + (−w), v/a = (1/a)v.

When the scalar field F is the real numbers R, the vector space is called a real vector space. When the scalar field is the complex numbers, it is called a complex vector space. These two cases are the ones used most often in engineering. The general definition of a vector space allows scalars to be elements of any fixed field F. The notion is then known as an F-vector spaces or a vector space over F. A field is, essentially, a set of numbers possessing addition, subtraction, multiplication and division operations.[nb 3] For example, rational numbers also form a field. In contrast to the intuition stemming from vectors in the plane and higher-dimensional cases, there is, in general vector spaces, no notion of nearness, angles or distances. To deal with such matters, particular types of vector spaces are introduced; see below.

11.1.4 Alternative formulations and elementary consequences

The requirement that vector addition and scalar multiplication be binary operations includes (by definition of binary operations) a property called closure: that u + v and av are in V for all a in F, and u, v in V. Some older sources mention these properties as separate axioms.[2] In the parlance of abstract algebra, the first four axioms can be subsumed by requiring the set of vectors to be an abelian group under addition. The remaining axioms give this group an F-module structure. In other words there is a ring homomorphism f from the field F into the endomorphism ring of the group of vectors. Then scalar multiplication av is defined as (f(a))(v).[3] There are a number of direct consequences of the vector space axioms. Some of them derive from elementary group theory, applied to the additive group of vectors: for example the zero vector 0 of V and the additive inverse −v of any vector v are unique. Other properties follow from the distributive law, for example av equals 0 if and only if a equals 0 or v equals 0.

11.2 History

Vector spaces stem from aﬃne geometry via the introduction of coordinates in the plane or three-dimensional space. Around 1636, Descartes and Fermat founded analytic geometry by equating solutions to an equation of two variables with points on a plane curve.[4] To achieve geometric solutions without using coordinates, Bolzano introduced, in 1804, certain operations on points, lines and planes, which are predecessors of vectors.[5] This work was made use of in the conception of barycentric coordinates by Möbius in 1827.[6] The foundation of the deﬁnition of vectors was Bellavitis' notion of the bipoint, an oriented segment one of whose ends is the origin and the other one a target. 110 CHAPTER 11. VECTOR SPACE

Vectors were reconsidered with the presentation of complex numbers by Argand and Hamilton and the inception of quaternions and biquaternions by the latter.[7] They are elements in R2, R4, and R8; treating them using linear combinations goes back to Laguerre in 1867, who also defined systems of linear equations. In 1857, Cayley introduced the matrix notation which allows for a harmonization and simplification of linear maps. Around the same time, Grassmann studied the barycentric calculus initiated by Möbius. He envisaged sets of abstract objects endowed with operations.[8] In his work, the concepts of linear independence and dimension, as well as scalar products, are present. Actually Grassmann’s 1844 work exceeds the framework of vector spaces, since his considering multiplication, too, led him to what are today called algebras. Peano was the first to give the modern definition of vector spaces and linear maps in 1888.[9] An important development of vector spaces is due to the construction of function spaces by Lebesgue. This was later formalized by Banach and Hilbert, around 1920.[10] At that time, algebra and the new field of functional analysis began to interact, notably with key concepts such as spaces of p-integrable functions and Hilbert spaces.[11] Vector spaces, including infinite-dimensional ones, then became a firmly established notion, and many mathematical branches started making use of this concept.

11.3 Examples

Main article: Examples of vector spaces

11.3.1 Coordinate spaces

Main article: Coordinate space

The most simple example of a vector space over a ﬁeld F is the ﬁeld itself, equipped with its standard addition and multiplication. More generally, a vector space can be composed of n-tuples (sequences of length n) of elements of F, such as

[12] (a1, a2, ..., an), where each ai is an element of F.

A vector space composed of all the n-tuples of a ﬁeld F is known as a coordinate space, usually denoted Fn. The case n = 1 is the above-mentioned simplest example, in which the ﬁeld F is also regarded as a vector space over itself. The case F = R and n = 2 was discussed in the introduction above.

11.3.2 Complex numbers and other ﬁeld extensions

The set of complex numbers C, i.e., numbers that can be written in the form x + iy for real numbers x and y where i is the imaginary unit, form a vector space over the reals with the usual addition and multiplication: (x + iy) + (a + ib) = (x + a) + i(y + b) and c ⋅ (x + iy) = (c ⋅ x) + i(c ⋅ y) for real numbers x, y, a, b and c. The various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic. In fact, the example of complex numbers is essentially the same (i.e., it is isomorphic) to the vector space of ordered pairs of real numbers mentioned above: if we think of the complex number x + i y as representing the ordered pair (x, y) in the complex plane then we see that the rules for sum and scalar product correspond exactly to those in the earlier example. More generally, field extensions provide another class of examples of vector spaces, particularly in algebra and algebraic number theory: a field F containing a smaller field E is an E-vector space, by the given multiplication and addition√ operations of F.[13] For example, the complex numbers are a vector space over R, and the field extension Q(i 5) is a vector space over Q.

11.3.3 Function spaces

Functions from any ﬁxed set Ω to a ﬁeld F also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions f and g is the function (f + g) given by 11.4. BASIS AND DIMENSION 111

(f + g)(w) = f(w) + g(w), and similarly for multiplication. Such function spaces occur in many geometric situations, when Ω is the real line or an interval, or other subsets of R. Many notions in topology and analysis, such as continuity, integrability or diﬀerentiability are well-behaved with respect to linearity: sums and scalar multiples of functions possessing such a property still have that property.[14] Therefore, the set of such functions are vector spaces. They are studied in greater detail using the methods of functional analysis, see below. Algebraic constraints also yield vector spaces: the vector space F[x] is given by polynomial functions:

n−1 n [15] f(x) = r0 + r1x + ... + rn₋₁x + rnx , where the coeﬃcients r0, ..., rn are in F.

11.3.4 Linear equations

Main articles: Linear equation, Linear diﬀerential equation and Systems of linear equations

Systems of homogeneous linear equations are closely tied to vector spaces.[16] For example, the solutions of

are given by triples with arbitrary a, b = a/2, and c = −5a/2. They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they are solutions, too. Matrices can be used to condense multiple linear equations as above into one vector equation, namely

Ax = 0, [ ] 1 3 1 where A = is the matrix containing the coeﬃcients of the given equations, x is the vector (a, b, c), Ax 4 2 2 denotes the matrix product, and 0 = (0, 0) is the zero vector. In a similar vein, the solutions of homogeneous linear diﬀerential equations form vector spaces. For example

f′′(x) + 2f′(x) + f(x) = 0 yields f(x) = a e−x + bx e−x, where a and b are arbitrary constants, and ex is the natural exponential function.

11.4 Basis and dimension

Main articles: Basis and Dimension Bases allow to represent vectors by a sequence of scalars called coordinates or components. A basis is a (finite or infinite) set B = {bi}i ∈ I of vectors bi, for convenience often indexed by some index set I, that spans the whole space and is linearly independent. “Spanning the whole space” means that any vector v can be expressed as a finite sum (called a linear combination) of the basis elements:

where the ak are scalars, called the coordinates (or the components) of the vector v with respect to the basis B, and bik (k = 1, ..., n) elements of B. Linear independence means that the coordinates ak are uniquely determined for any vector in the vector space. n For example, the coordinate vectors e1 = (1, 0, ..., 0), e2 = (0, 1, 0, ..., 0), to en = (0, 0, ..., 0, 1), form a basis of F , called the standard basis, since any vector (x1, x2, ..., xn) can be uniquely expressed as a linear combination of these vectors:

(x1, x2, ..., xn) = x1(1, 0, ..., 0) + x2(0, 1, 0, ..., 0) + ... + xn(0, ..., 0, 1) = x1e1 + x2e2 + ... + xnen. 112 CHAPTER 11. VECTOR SPACE

2 2 A vector v in R (blue) expressed in terms of diﬀerent bases: using the standard basis of R v = xe1 + ye2 (black), and using a diﬀerent, non-orthogonal basis: v = f1 + f2 (red).

The corresponding coordinates x1, x2, ..., xn are just the Cartesian coordinates of the vector. Every vector space has a basis. This follows from Zorn’s lemma, an equivalent formulation of the Axiom of Choice.[17] Given the other axioms of Zermelo–Fraenkel set theory, the existence of bases is equivalent to the axiom of choice.[18] The ultrafilter lemma, which is weaker than the axiom of choice, implies that all bases of a given vector space have the same number of elements, or cardinality (cf. Dimension theorem for vector spaces).[19] It is called the dimension of the vector space, denoted dim V. If the space is spanned by finitely many vectors, the above statements can be proven without such fundamental input from set theory.[20] The dimension of the coordinate space Fn is n, by the basis exhibited above. The dimension of the polynomial ring F[x] introduced above is countably infinite, a basis is given by 1, x, x2, ... A fortiori, the dimension of more general function spaces, such as the space of functions on some (bounded or unbounded) interval, is infinite.[nb 4] Under suitable regularity assumptions on the coefficients involved, the dimension of the solution space of a homogeneous ordinary differential equation equals the degree of the equation.[21] For example, the solution space for the above equation is generated by e−x and xe−x. These two functions are linearly independent over R, so the dimension of this space is two, as is the degree of the equation. A field extension over the rationals Q can be thought of as a vector space over Q (by defining vector addition as field addition, defining scalar multiplication as field multiplication by elements of Q, and otherwise ignoring the field multiplication). The dimension (or degree) of the field extension Q(α) over Q depends on α. If α satisfies some 11.5. LINEAR MAPS AND MATRICES 113 polynomial equation

n n−1 qnα + qn₋₁α + ... + q0 = 0, with rational coeﬃcients qn, ..., q0.

("α is algebraic"), the dimension is finite. More precisely, it equals the degree of the minimal polynomial having α as a root.[22] For example, the complex numbers C are a two-dimensional real vector space, generated by 1 and the imaginary unit i. The latter satisfies i2 + 1 = 0, an equation of degree two. Thus, C is a two-dimensional R-vector space (and, as any field, one-dimensional as a vector space over itself, C). If α is not algebraic, the dimension of Q(α) over Q is infinite. For instance, for α = π there is no such equation, in other words π is transcendental.[23]

11.5 Linear maps and matrices

Main article: Linear map

The relation of two vector spaces can be expressed by linear map or linear transformation. They are functions that reﬂect the vector space structure—i.e., they preserve sums and scalar multiplication:

f(x + y) = f(x) + f(y) and f(a · x) = a · f(x) for all x and y in V, all a in F.[24]

An isomorphism is a linear map f : V → W such that there exists an inverse map g : W → V, which is a map such that the two possible compositions f ∘ g : W → W and g ∘ f : V → V are identity maps. Equivalently, f is both one-to-one (injective) and onto (surjective).[25] If there exists an isomorphism between V and W, the two spaces are said to be isomorphic; they are then essentially identical as vector spaces, since all identities holding in V are, via f, transported to similar ones in W, and vice versa via g.

Describing an arrow vector v by its coordinates x and y yields an isomorphism of vector spaces. 114 CHAPTER 11. VECTOR SPACE

For example, the “arrows in the plane” and “ordered pairs of numbers” vector spaces in the introduction are isomorphic: a planar arrow v departing at the origin of some (fixed) coordinate system can be expressed as an ordered pair by considering the x- and y-component of the arrow, as shown in the image at the right. Conversely, given a pair (x, y), the arrow going by x to the right (or to the left, if x is negative), and y up (down, if y is negative) turns back the arrow v. Linear maps V → W between two vector spaces form a vector space HomF(V, W), also denoted L(V, W).[26] The space of linear maps from V to F is called the dual vector space, denoted V∗.[27] Via the injective natural map V → V∗∗, any vector space can be embedded into its bidual; the map is an isomorphism if and only if the space is finite-dimensional.[28] Once a basis of V is chosen, linear maps f : V → W are completely determined by specifying the images of the basis vectors, because any element of V is expressed uniquely as a linear combination of them.[29] If dim V = dim W, a 1-to-1 correspondence between fixed bases of V and W gives rise to a linear map that maps any basis element of V to the corresponding basis element of W. It is an isomorphism, by its very definition.[30] Therefore, two vector spaces are isomorphic if their dimensions agree and vice versa. Another way to express this is that any vector space is completely classified (up to isomorphism) by its dimension, a single number. In particular, any n-dimensional F-vector space V is isomorphic to Fn. There is, however, no “canonical” or preferred isomorphism; actually an isomorphism φ : Fn → V is equivalent to the choice of a basis of V, by mapping the standard basis of Fn to V, via φ. The freedom of choosing a convenient basis is particularly useful in the infinite-dimensional context, see below.

11.5.1 Matrices

Main articles: Matrix and Determinant Matrices are a useful notion to encode linear maps.[31] They are written as a rectangular array of scalars as in the

ai,j n columns j changes m rows . . a1,1 a1,2 a1,3 . i c . . a2,1 . h a2,2 a2,3 a n . . a3,1 a3,2 a3,3 . g e . . . . s ......

A typical matrix

image at the right. Any m-by-n matrix A gives rise to a linear map from Fn to Fm, by the following 11.5. LINEAR MAPS AND MATRICES 115

(∑ ∑ ∑ ) ∑ ··· 7→ n n ··· n x = (x1, x2, , xn) j=1 a1jxj, j=1 a2jxj, , j=1 amjxj , where denotes summation,

or, using the matrix multiplication of the matrix A with the coordinate vector x:

x ↦ Ax.

Moreover, after choosing bases of V and W, any linear map f : V → W is uniquely represented by a matrix via this assignment.[32]

The volume of this parallelepiped is the absolute value of the determinant of the 3-by-3 matrix formed by the vectors r1, r2, and r3.

The determinant det (A) of a square matrix A is a scalar that tells whether the associated map is an isomorphism or not: to be so it is suﬃcient and necessary that the determinant is nonzero.[33] The linear transformation of Rn corresponding to a real n-by-n matrix is orientation preserving if and only if its determinant is positive.

11.5.2 Eigenvalues and eigenvectors

Main article: Eigenvalues and eigenvectors

Endomorphisms, linear maps f : V → V, are particularly important since in this case vectors v can be compared with their image under f, f(v). Any nonzero vector v satisfying λv = f(v), where λ is a scalar, is called an eigenvector of f with eigenvalue λ.[nb 5][34] Equivalently, v is an element of the kernel of the diﬀerence f − λ · Id (where Id is the identity map V → V). If V is ﬁnite-dimensional, this can be rephrased using determinants: f having eigenvalue λ is equivalent to

det(f − λ · Id) = 0. 116 CHAPTER 11. VECTOR SPACE

By spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function in λ, called the characteristic polynomial of f.[35] If the field F is large enough to contain a zero of this polynomial (which automatically happens for F algebraically closed, such as F = C) any linear map has at least one eigenvector. The vector space V may or may not possess an eigenbasis, a basis consisting of eigenvectors. This phenomenon is governed by the Jordan canonical form of the map.[nb 6] The set of all eigenvectors corresponding to a particular eigenvalue of f forms a vector space known as the eigenspace corresponding to the eigenvalue (and f) in question. To achieve the spectral theorem, the corresponding statement in the infinite-dimensional case, the machinery of functional analysis is needed, see below.

11.6 Basic constructions

In addition to the above concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones. In addition to the deﬁnitions given below, they are also characterized by universal properties, which determine an object X by specifying the linear maps from X to any other vector space.

11.6.1 Subspaces and quotient spaces

Main articles: Linear subspace and Quotient vector space A nonempty subset W of a vector space V that is closed under addition and scalar multiplication (and therefore

A line passing through the origin (blue, thick) in R3 is a linear subspace. It is the intersection of two planes (green and yellow).

contains the 0-vector of V) is called a subspace of V.[36] Subspaces of V are vector spaces (over the same ﬁeld) in their own right. The intersection of all subspaces containing a given set S of vectors is called its span, and it is the smallest subspace of V containing the set S. Expressed in terms of elements, the span is the subspace consisting of all the linear combinations of elements of S.[37] The counterpart to subspaces are quotient vector spaces.[38] Given any subspace W ⊂ V, the quotient space V/W ("V modulo W") is deﬁned as follows: as a set, it consists of v + W = {v + w : w ∈ W}, where v is an arbitrary vector in V. The sum of two such elements v1 + W and v2 + W is (v1 + v2) + W, and scalar multiplication is given by a ·(v + 11.6. BASIC CONSTRUCTIONS 117

W) = (a · v) + W. The key point in this definition is that v1 + W = v2 + W if and only if the difference of v1 and v2 lies in W.[nb 7] This way, the quotient space “forgets” information that is contained in the subspace W. The kernel ker(f) of a linear map f : V → W consists of vectors v that are mapped to 0 in W.[39] Both kernel and image im(f) = {f(v): v ∈ V} are subspaces of V and W, respectively.[40] The existence of kernels and images is part of the statement that the category of vector spaces (over a fixed field F) is an abelian category, i.e. a corpus of mathematical objects and structure-preserving maps between them (a category) that behaves much like the category of abelian groups.[41] Because of this, many statements such as the first isomorphism theorem (also called rank–nullity theorem in matrix-related terms)

V / ker(f) ≡ im(f). and the second and third isomorphism theorem can be formulated and proven in a way very similar to the corresponding statements for groups. An important example is the kernel of a linear map x ↦ Ax for some ﬁxed matrix A, as above. The kernel of this map is the subspace of vectors x such that Ax = 0, which is precisely the set of solutions to the system of homogeneous linear equations belonging to A. This concept also extends to linear diﬀerential equations

df d2f ··· dnf a0f + a1 dx + a2 dx2 + + an dxn = 0 , where the coeﬃcients ai are functions in x, too.

In the corresponding map

∑n dif f 7→ D(f) = a i dxi i=0

the derivatives of the function f appear linearly (as opposed to f′′(x)2, for example). Since differentiation is a linear procedure (i.e., (f + g)′ = f′ + g ′ and (c·f)′ = c·f′ for a constant c) this assignment is linear, called a linear differential operator. In particular, the solutions to the differential equation D(f) = 0 form a vector space (over R or C).

11.6.2 Direct product and direct sum

Main articles: Direct product and Direct sum of modules

The direct product of vector spaces and the direct sum of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space. ∏ The direct product i∈I Vi of a family of vector spaces Vi consists of the set of all tuples (vi)i ∈ I, which specify for [42] each index i in some index set I an element vi of Vi. Addition and scalar multiplication is performed⨿ componen- ⊕ twise. A variant of this construction is the direct sum i∈I Vi (also called coproduct and denoted i∈I Vi ), where only tuples with finitely many nonzero vectors are allowed. If the index set I is finite, the two constructions agree, but in general they are different.

11.6.3 Tensor product

Main article: Tensor product of vector spaces

The tensor product V ⊗FW, or simply V ⊗ W, of two vector spaces V and W is one of the central notions of multilinear algebra which deals with extending notions such as linear maps to several variables. A map g : V × W → X is called bilinear if g is linear in both variables v and w. That is to say, for fixed w the map v ↦ g(v, w) is linear in the sense above and likewise for fixed v. The tensor product is a particular vector space that is a universal recipient of bilinear maps g, as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called tensors

v1 ⊗ w1 + v2 ⊗ w2 + ... + vn ⊗ wn, 118 CHAPTER 11. VECTOR SPACE

subject to the rules

a ·(v ⊗ w) = (a · v) ⊗ w = v ⊗ (a · w), where a is a scalar,

(v1 + v2) ⊗ w = v1 ⊗ w + v2 ⊗ w, and [43] v ⊗ (w1 + w2) = v ⊗ w1 + v ⊗ w2.

Commutative diagram depicting the universal property of the tensor product.

These rules ensure that the map f from the V × W to V ⊗ W that maps a tuple (v, w) to v ⊗ w is bilinear. The universality states that given any vector space X and any bilinear map g : V × W → X, there exists a unique map u, shown in the diagram with a dotted arrow, whose composition with f equals g: u(v ⊗ w) = g(v, w).[44] This is called the universal property of the tensor product, an instance of the method—much used in advanced abstract algebra—to indirectly deﬁne objects by specifying maps from or to this object.

11.7 Vector spaces with additional structure

From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space is characterized, up to isomorphism, by its dimension. However, vector spaces per se do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions converges to another function. Likewise, linear algebra is not adapted to deal with infinite series, since the addition operation allows only finitely many terms to be added. Therefore, the needs of functional analysis require considering additional structures. A vector space may be given a partial order ≤, under which some vectors can be compared.[45] For example, n- dimensional real space Rn can be ordered by comparing its vectors componentwise. Ordered vector spaces, for example Riesz spaces, are fundamental to Lebesgue integration, which relies on the ability to express a function as a difference of two positive functions

f = f+ − f−,

where f+ denotes the positive part of f and f− the negative part.[46]

11.7.1 Normed vector spaces and inner product spaces

Main articles: Normed vector space and Inner product space 11.7. VECTOR SPACES WITH ADDITIONAL STRUCTURE 119

“Measuring” vectors is done by specifying a norm, a datum which measures lengths of vectors, or by an inner product, which measures angles between vectors. Norms and inner products are denoted |v| and ⟨v, w⟩ , respectively. The √datum of an inner product entails that lengths of vectors can be deﬁned too, by deﬁning the associated norm |v| := ⟨v, v⟩ . Vector spaces endowed with such data are known as normed vector spaces and inner product spaces, respectively.[47] Coordinate space Fn can be equipped with the standard dot product:

⟨x, y⟩ = x · y = x1y1 + ··· + xnyn.

In R2, this reﬂects the common notion of the angle between two vectors x and y, by the law of cosines:

x · y = cos (∠(x, y)) · |x| · |y|.

Because of this, two vectors satisfying ⟨x, y⟩ = 0 are called orthogonal. An important variant of the standard dot product is used in Minkowski space: R4 endowed with the Lorentz product

[48] ⟨x|y⟩ = x1y1 + x2y2 + x3y3 − x4y4.

In contrast to the standard dot product, it is not positive deﬁnite: ⟨x|x⟩ also takes negative values, for example for x = (0, 0, 0, 1) . Singling out the fourth coordinate—corresponding to time, as opposed to three space-dimensions— makes it useful for the mathematical treatment of special relativity.

11.7.2 Topological vector spaces

Main article: Topological vector space

Convergence questions are treated by considering vector spaces V carrying a compatible topology, a structure that allows one to talk about elements being close to each other.[49][50] Compatible here means that addition and scalar multiplication have to be continuous maps. Roughly, if x and y in V, and a in F vary by a bounded amount, then so do x + y and ax.[nb 8] To make sense of specifying the amount a scalar changes, the ﬁeld F also has to carry a topology in this context; a common choice are the reals or the complex numbers. In such topological vector spaces one can consider series of vectors. The inﬁnite sum

∑∞ fi i=0 denotes the limit of the corresponding finite partial sums of the sequence (fi)i∈N of elements of V. For example, the fi could be (real or complex) functions belonging to some function space V, in which case the series is a function series. The mode of convergence of the series depends on the topology imposed on the function space. In such cases, pointwise convergence and uniform convergence are two prominent examples. A way to ensure the existence of limits of certain infinite series is to restrict attention to spaces where any Cauchy sequence has a limit; such a vector space is called complete. Roughly, a vector space is complete provided that it contains all necessary limits. For example, the vector space of polynomials on the unit interval [0,1], equipped with the topology of uniform convergence is not complete because any continuous function on [0,1] can be uniformly approximated by a sequence of polynomials, by the Weierstrass approximation theorem.[51] In contrast, the space of all continuous functions on [0,1] with the same topology is complete.[52] A norm gives rise to a topology by defining that a sequence of vectors vn converges to v if and only if

limn→∞|vn − v| = 0.

Banach and Hilbert spaces are complete topological vector spaces whose topologies are given, respectively, by a norm and an inner product. Their study—a key piece of functional analysis—focusses on inﬁnite-dimensional vector spaces, 120 CHAPTER 11. VECTOR SPACE

2 1

∞

2 Unit “spheres” in R consist of plane vectors of norm√ 1. Depicted are the unit spheres in different p-norms, for p = 1, 2, and ∞. The bigger diamond depicts points of 1-norm equal to 2 . since all norms on finite-dimensional topological vector spaces give rise to the same notion of convergence.[53] The image at the right shows the equivalence of the 1-norm and ∞-norm on R2: as the unit “balls” enclose each other, a sequence converges to zero in one norm if and only if it so does in the other norm. In the infinite-dimensional case, however, there will generally be inequivalent topologies, which makes the study of topological vector spaces richer than that of vector spaces without additional data. From a conceptual point of view, all notions related to topological vector spaces should match the topology. For example, instead of considering all linear maps (also called functionals) V → W, maps between topological vector spaces are required to be continuous.[54] In particular, the (topological) dual space V∗ consists of continuous functionals V → R (or to C). The fundamental Hahn–Banach theorem is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.[55]

Banach spaces

Main article: Banach space 11.7. VECTOR SPACES WITH ADDITIONAL STRUCTURE 121

Banach spaces, introduced by Stefan Banach, are complete normed vector spaces.[56] A first example is the vector p space ℓ consisting of infinite vectors with real entries x = (x1, x2, ...) whose p-norm (1 ≤ p ≤ ∞) given by ∑ | | | |p 1/p | | | | x p := ( i xi ) for p < ∞ and x ∞ := supi xi is finite. The topologies on the infinite-dimensional space ℓ p are inequivalent for different p. E.g. the sequence of vectors xn = (2−n, 2−n, ..., 2−n, 0, 0, ...), i.e. the first 2n components are 2−n, the following ones are 0, converges to the zero vector for p = ∞, but does not for p = 1:

∑ n | | −n −n → | | 2 −n n · −n xn ∞ = sup(2 , 0) = 2 0 , but xn 1 = i=1 2 = 2 2 = 1.

More generally than sequences of real numbers, functions f: Ω → R are endowed with a norm that replaces the above sum by the Lebesgue integral

(∫ )1/p p |f|p := |f(x)| dx . Ω The space of integrable functions on a given domain Ω (for example an interval) satisfying |f|p < ∞, and equipped with this norm are called Lebesgue spaces, denoted Lp(Ω).[nb 9] These spaces are complete.[57] (If one uses the Riemann integral instead, the space is not complete, which may be seen as a justiﬁcation for Lebesgue’s integration theory.[nb 10]) Concretely this means that for any sequence of Lebesgue-integrable functions f1, f2, ... with |fn|p < ∞, satisfying the condition

∫ p lim |fk(x) − fn(x)| dx = 0 →∞ k, n Ω there exists a function f(x) belonging to the vector space Lp(Ω) such that

∫ p lim |f(x) − fk(x)| dx = 0. →∞ k Ω Imposing boundedness conditions not only on the function, but also on its derivatives leads to Sobolev spaces.[58]

Hilbert spaces

Main article: Hilbert space Complete inner product spaces are known as Hilbert spaces, in honor of David Hilbert.[59] The Hilbert space L2(Ω),

The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by ﬁnite sum of sine functions (red).

with inner product given by

∫ ⟨f , g⟩ = f(x)g(x) dx, Ω 122 CHAPTER 11. VECTOR SPACE

where g(x) denotes the complex conjugate of g(x),[60][nb 11] is a key case. By definition, in a Hilbert space any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions fn with desirable properties that approximates a given limit function, is equally crucial. Early analysis, in the guise of the Taylor approximation, established an approximation of differentiable functions f by polynomials.[61] By the Stone–Weierstrass theorem, every continuous function on [a, b] can be approximated as closely as desired by a polynomial.[62] A similar approximation technique by trigonometric functions is commonly called Fourier expansion, and is much applied in engineering, see below. More generally, and more conceptually, the theorem yields a simple description of what “basic functions”, or, in abstract Hilbert spaces, what basic vectors suffice to generate a Hilbert space H, in the sense that the closure of their span (i.e., finite linear combinations and limits of those) is the whole space. Such a set of functions is called a basis of H, its cardinality is known as the Hilbert space dimension.[nb 12] Not only does the theorem exhibit suitable basis functions as sufficient for approximation purposes, but together with the Gram-Schmidt process, it enables one to construct a basis of orthogonal vectors.[63] Such orthogonal bases are the Hilbert space generalization of the coordinate axes in finite-dimensional Euclidean space. The solutions to various differential equations can be interpreted in terms of Hilbert spaces. For example, a great many fields in physics and engineering lead to such equations and frequently solutions with particular physical properties are used as basis functions, often orthogonal.[64] As an example from physics, the time-dependent Schrödinger equation in quantum mechanics describes the change of physical properties in time by means of a partial differential equation, whose solutions are called wavefunctions.[65] Definite values for physical properties such as energy, or momentum, correspond to eigenvalues of a certain (linear) differential operator and the associated wavefunctions are called eigenstates. The spectral theorem decomposes a linear compact operator acting on functions in terms of these eigenfunctions and their eigenvalues.[66]

11.7.3 Algebras over ﬁelds

Main articles: Algebra over a field and Lie algebra General vector spaces do not possess a multiplication between vectors. A vector space equipped with an additional bilinear operator defining the multiplication of two vectors is an algebra over a field.[67] Many algebras stem from functions on some geometrical object: since functions with values in a given field can be multiplied pointwise, these entities form algebras. The Stone–Weierstrass theorem mentioned above, for example, relies on Banach algebras which are both Banach spaces and algebras. Commutative algebra makes great use of rings of polynomials in one or several variables, introduced above. Their multiplication is both commutative and associative. These rings and their quotients form the basis of algebraic geometry, because they are rings of functions of algebraic geometric objects.[68] Another crucial example are Lie algebras, which are neither commutative nor associative, but the failure to be so is limited by the constraints ([x, y] denotes the product of x and y):

• [x, y] = −[y, x](anticommutativity), and

• [x,[y, z]] + [y,[z, x]] + [z,[x, y]] = 0 (Jacobi identity).[69]

Examples include the vector space of n-by-n matrices, with [x, y] = xy − yx, the commutator of two matrices, and R3, endowed with the cross product. The tensor algebra T(V) is a formal way of adding products to any vector space V to obtain an algebra.[70] As a vector space, it is spanned by symbols, called simple tensors

v1 ⊗ v2 ⊗ ... ⊗ vn, where the degree n varies.

The multiplication is given by concatenating such symbols, imposing the distributive law under addition, and requiring that scalar multiplication commute with the tensor product ⊗, much the same way as with the tensor product of two vector spaces introduced above. In general, there are no relations between v1 ⊗ v2 and v2 ⊗ v1. Forcing two such [71] elements to be equal leads to the symmetric algebra, whereas forcing v1 ⊗ v2 = − v2 ⊗ v1 yields the exterior algebra. When a ﬁeld, F is explicitly stated, a common term used is F-algebra. 11.8. APPLICATIONS 123

A hyperbola, given by the equation x y = 1. The coordinate ring of functions on this hyperbola is given by R[x, y]/(x · y − 1), an inﬁnite-dimensional vector space over R.

11.8 Applications

Vector spaces have manifold applications as they occur in many circumstances, namely wherever functions with values in some ﬁeld are involved. They provide a framework to deal with analytical and geometrical problems, or are used in the Fourier transform. This list is not exhaustive: many more applications exist, for example in optimization. The minimax theorem of game theory stating the existence of a unique payoﬀ when all players play optimally can be formulated and proven using vector spaces methods.[72] Representation theory fruitfully transfers the good understanding of linear algebra and vector spaces to other mathematical domains such as group theory.[73]

11.8.1 Distributions

Main article: Distribution

A distribution (or generalized function) is a linear map assigning a number to each “test” function, typically a smooth function with compact support, in a continuous way: in the above terminology the space of distributions is the (continuous) dual of the test function space.[74] The latter space is endowed with a topology that takes into account not 124 CHAPTER 11. VECTOR SPACE

only f itself, but also all its higher derivatives. A standard example is the result of integrating a test function f over some domain Ω:

∫ I(f) = f(x) dx. Ω When Ω = {p}, the set consisting of a single point, this reduces to the Dirac distribution, denoted by δ, which associates to a test function f its value at the p: δ(f) = f(p). Distributions are a powerful instrument to solve differential equations. Since all standard analytic notions such as derivatives are linear, they extend naturally to the space of distributions. Therefore the equation in question can be transferred to a distribution space, which is bigger than the underlying function space, so that more flexible methods are available for solving the equation. For example, Green’s functions and fundamental solutions are usually distributions rather than proper functions, and can then be used to find solutions of the equation with prescribed boundary conditions. The found solution can then in some cases be proven to be actually a true function, and a solution to the original equation (e.g., using the Lax–Milgram theorem, a consequence of the Riesz representation theorem).[75]

11.8.2 Fourier analysis

Main article: Fourier analysis Resolving a periodic function into a sum of trigonometric functions forms a Fourier series, a technique much used

The heat equation describes the dissipation of physical properties over time, such as the decline of the temperature of a hot body placed in a colder environment (yellow depicts colder regions than red).

in physics and engineering.[nb 13][76] The underlying vector space is usually the Hilbert space L2(0, 2π), for which the functions sin mx and cos mx (m an integer) form an orthogonal basis.[77] The Fourier expansion of an L2 function f is

∞ a ∑ 0 + [a cos (mx) + b sin (mx)] . 2 m m m=1 The coeﬃcients am and bm are called Fourier coeﬃcients of f, and are calculated by the formulas[78] 11.8. APPLICATIONS 125

∫ ∫ 1 2π 1 2π am = π 0 f(t) cos(mt) dt , bm = π 0 f(t) sin(mt) dt. In physical terms the function is represented as a superposition of sine waves and the coefficients give information about the function’s frequency spectrum.[79] A complex-number form of Fourier series is also commonly used.[78] The concrete formulae above are consequences of a more general mathematical duality called Pontryagin duality.[80] Applied to the group R, it yields the classical Fourier transform; an application in physics are reciprocal lattices, where the underlying group is a finite-dimensional real vector space endowed with the additional datum of a lattice encoding positions of atoms in crystals.[81] Fourier series are used to solve boundary value problems in partial differential equations.[82] In 1822, Fourier first used this technique to solve the heat equation.[83] A discrete version of the Fourier series can be used in sampling applications where the function value is known only at a finite number of equally spaced points. In this case the Fourier series is finite and its value is equal to the sampled values at all points.[84] The set of coefficients is known as the discrete Fourier transform (DFT) of the given sample sequence. The DFT is one of the key tools of digital signal processing, a field whose applications include radar, speech encoding, image compression.[85] The JPEG image format is an application of the closely related discrete cosine transform.[86] The fast Fourier transform is an algorithm for rapidly computing the discrete Fourier transform.[87] It is used not only for calculating the Fourier coefficients but, using the convolution theorem, also for computing the convolution of two finite sequences.[88] They in turn are applied in digital filters[89] and as a rapid multiplication algorithm for polynomials and large integers (Schönhage-Strassen algorithm).[90][91]

11.8.3 Diﬀerential geometry

Main article: Tangent space The tangent plane to a surface at a point is naturally a vector space whose origin is identiﬁed with the point of

The tangent space to the 2-sphere at some point is the infinite plane touching the sphere in this point. contact. The tangent plane is the best linear approximation, or linearization, of a surface at a point.[nb 14] Even in a three-dimensional Euclidean space, there is typically no natural way to prescribe a basis of the tangent plane, and so it is conceived of as an abstract vector space rather than a real coordinate space. The tangent space is the generalization to higher-dimensional differentiable manifolds.[92] Riemannian manifolds are manifolds whose tangent spaces are endowed with a suitable inner product.[93] Derived therefrom, the Riemann curvature tensor encodes all curvatures of a manifold in one object, which finds applications 126 CHAPTER 11. VECTOR SPACE in general relativity, for example, where the Einstein curvature tensor describes the matter and energy content of space-time.[94][95] The tangent space of a Lie group can be given naturally the structure of a Lie algebra and can be used to classify compact Lie groups.[96]

11.9 Generalizations

11.9.1 Vector bundles

Main articles: Vector bundle and Tangent bundle A vector bundle is a family of vector spaces parametrized continuously by a topological space X.[92] More precisely,

1 S

1 U x R U

A Möbius strip. Locally, it looks like U × R. a vector bundle over X is a topological space E equipped with a continuous map

π : E → X such that for every x in X, the ﬁber π−1(x) is a vector space. The case dim V = 1 is called a line bundle. For any vector space V, the projection X × V → X makes the product X × V into a “trivial” vector bundle. Vector bundles over X are required to be locally a product of X and some (ﬁxed) vector space V: for every x in X, there is a neighborhood U of x such that the restriction of π to π−1(U) is isomorphic[nb 15] to the trivial bundle U × V → U. Despite their locally trivial character, vector bundles may (depending on the shape of the underlying space X) be “twisted” in the large 11.10. SEE ALSO 127

(i.e., the bundle need not be (globally isomorphic to) the trivial bundle X × V). For example, the Möbius strip can be seen as a line bundle over the circle S1 (by identifying open intervals with the real line). It is, however, different from the cylinder S1 × R, because the latter is orientable whereas the former is not.[97] Properties of certain vector bundles provide information about the underlying topological space. For example, the tangent bundle consists of the collection of tangent spaces parametrized by the points of a differentiable manifold. The tangent bundle of the circle S1 is globally isomorphic to S1 × R, since there is a global nonzero vector field on S1.[nb 16] In contrast, by the hairy ball theorem, there is no (tangent) vector field on the 2-sphere S2 which is everywhere nonzero.[98] K-theory studies the isomorphism classes of all vector bundles over some topological space.[99] In addition to deepening topological and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional real division algebras: R, C, the quaternions H and the octonions. The cotangent bundle of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the cotangent space. Sections of that bundle are known as differential one-forms.

11.9.2 Modules

Main article: Module

Modules are to rings what vector spaces are to fields. The very same axioms, applied to a ring R instead of a field F yield modules.[100] The theory of modules, compared to that of vector spaces, is complicated by the presence of ring elements that do not have multiplicative inverses. For example, modules need not have bases, as the Z-module (i.e., abelian group) Z/2Z shows; those modules that do (including all vector spaces) are known as free modules. Nevertheless, a vector space can be compactly defined as a module over a ring which is a field with the elements being called vectors. Some authors use the term vector space to mean modules over a division ring.[101] The algebro- geometric interpretation of commutative rings via their spectrum allows the development of concepts such as locally free modules, the algebraic counterpart to vector bundles.

11.9.3 Aﬃne and projective spaces

Main articles: Affine space and Projective space Roughly, affine spaces are vector spaces whose origins are not specified.[102] More precisely, an affine space is a set with a free transitive vector space action. In particular, a vector space is an affine space over itself, by the map

V × V → V,(v, a) ↦ a + v.

If W is a vector space, then an aﬃne subspace is a subset of W obtained by translating a linear subspace V by a ﬁxed vector x ∈ W; this space is denoted by x + V (it is a coset of V in W) and consists of all vectors of the form x + v for v ∈ V. An important example is the space of solutions of a system of inhomogeneous linear equations

Ax = b

generalizing the homogeneous case b = 0 above.[103] The space of solutions is the affine subspace x + V where x is a particular solution of the equation, and V is the space of solutions of the homogeneous equation (the nullspace of A). The set of one-dimensional subspaces of a fixed finite-dimensional vector space V is known as projective space; it may be used to formalize the idea of parallel lines intersecting at infinity.[104] Grassmannians and flag manifolds generalize this by parametrizing linear subspaces of fixed dimension k and flags of subspaces, respectively.

11.10 See also

• Vector (mathematics and physics), for a list of various kinds of vectors

11.11 Notes

[1] It is also common, especially in physics, to denote vectors with an arrow on top: ⃗v . 128 CHAPTER 11. VECTOR SPACE

An aﬃne plane (light blue) in R3. It is a two-dimensional subspace shifted by a vector x (red).

[2] This axiom refers to two diﬀerent operations: scalar multiplication: bv; and ﬁeld multiplication: ab. It does not assert the associativity of either operation. More formally, scalar multiplication is the semigroup action of the scalars on the vector space. Combined with the axiom of the identity element of scalar multiplication, it is a monoid action.

[3] Some authors (such as Brown 1991) restrict attention to the ﬁelds R or C, but most of the theory is unchanged for an arbitrary ﬁeld.

[4] The indicator functions of intervals (of which there are inﬁnitely many) are linearly independent, for example.

[5] The nomenclature derives from German "eigen", which means own or proper.

[6] Roman 2005, ch. 8, p. 140. See also Jordan–Chevalley decomposition.

[7] Some authors (such as Roman 2005) choose to start with this equivalence relation and derive the concrete shape of V/W from this.

[8] This requirement implies that the topology gives rise to a uniform structure, Bourbaki 1989, ch. II

[9] The triangle inequality for |−|p is provided by the Minkowski inequality. For technical reasons, in the context of functions one has to identify functions that agree almost everywhere to get a norm, and not only a seminorm.

[10] “Many functions in L2 of Lebesgue measure, being unbounded, cannot be integrated with the classical Riemann integral. So spaces of Riemann integrable functions would not be complete in the L2 norm, and the orthogonal decomposition would not apply to them. This shows one of the advantages of Lebesgue integration.”, Dudley 1989, §5.3, p. 125

[11] For p ≠2, Lp(Ω) is not a Hilbert space.

[12] A basis of a Hilbert space is not the same thing as a basis in the sense of linear algebra above. For distinction, the latter is then called a Hamel basis.

[13] Although the Fourier series is periodic, the technique can be applied to any L2 function on an interval by considering the function to be continued periodically outside the interval. See Kreyszig 1988, p. 601 11.12. FOOTNOTES 129

[14] That is to say (BSE-3 2001), the plane passing through the point of contact P such that the distance from a point P1 on the surface to the plane is inﬁnitesimally small compared to the distance from P1 to P in the limit as P1 approaches P along the surface.

[15] That is, there is a homeomorphism from π−1(U) to V × U which restricts to linear isomorphisms between ﬁbers.

[16] A line bundle, such as the tangent bundle of S1 is trivial if and only if there is a section that vanishes nowhere, see Husemoller 1994, Corollary 8.3. The sections of the tangent bundle are just vector ﬁelds.

11.12 Footnotes

[1] Roman 2005, ch. 1, p. 27

[2] van der Waerden 1993, Ch. 19

[3] Bourbaki 1998, §II.1.1. Bourbaki calls the group homomorphisms f(a) homotheties.

[4] Bourbaki 1969, ch. “Algèbre linéaire et algèbre multilinéaire”, pp. 78–91

[5] Bolzano 1804

[6] Möbius 1827

[7] Hamilton 1853

[8] Grassmann 2000

[9] Peano 1888, ch. IX

[10] Banach 1922

[11] Dorier 1995, Moore 1995

[12] Lang 1987, ch. I.1

[13] Lang 2002, ch. V.1

[14] e.g. Lang 1993, ch. XII.3., p. 335

[15] Lang 1987, ch. IX.1

[16] Lang 1987, ch. VI.3.

[17] Roman 2005, Theorem 1.9, p. 43

[18] Blass 1984

[19] Halpern 1966, pp. 670–673

[20] Artin 1991, Theorem 3.3.13

[21] Braun 1993, Th. 3.4.5, p. 291

[22] Stewart 1975, Proposition 4.3, p. 52

[23] Stewart 1975, Theorem 6.5, p. 74

[24] Roman 2005, ch. 2, p. 45

[25] Lang 1987, ch. IV.4, Corollary, p. 106

[26] Lang 1987, Example IV.2.6

[27] Lang 1987, ch. VI.6

[28] Halmos 1974, p. 28, Ex. 9

[29] Lang 1987, Theorem IV.2.1, p. 95

[30] Roman 2005, Th. 2.5 and 2.6, p. 49 130 CHAPTER 11. VECTOR SPACE

[31] Lang 1987, ch. V.1

[32] Lang 1987, ch. V.3., Corollary, p. 106

[33] Lang 1987, Theorem VII.9.8, p. 198

[34] Roman 2005, ch. 8, p. 135–156

[35] Lang 1987, ch. IX.4

[36] Roman 2005, ch. 1, p. 29

[37] Roman 2005, ch. 1, p. 35

[38] Roman 2005, ch. 3, p. 64

[39] Lang 1987, ch. IV.3.

[40] Roman 2005, ch. 2, p. 48

[41] Mac Lane 1998

[42] Roman 2005, ch. 1, pp. 31–32

[43] Lang 2002, ch. XVI.1

[44] Roman 2005, Th. 14.3. See also Yoneda lemma.

[45] Schaefer & Wolﬀ 1999, pp. 204–205

[46] Bourbaki 2004, ch. 2, p. 48

[47] Roman 2005, ch. 9

[48] Naber 2003, ch. 1.2

[49] Treves 1967

[50] Bourbaki 1987

[51] Kreyszig 1989, §4.11-5

[52] Kreyszig 1989, §1.5-5

[53] Choquet 1966, Proposition III.7.2

[54] Treves 1967, p. 34–36

[55] Lang 1983, Cor. 4.1.2, p. 69

[56] Treves 1967, ch. 11

[57] Treves 1967, Theorem 11.2, p. 102

[58] Evans 1998, ch. 5

[59] Treves 1967, ch. 12

[60] Dennery 1996, p.190

[61] Lang 1993, Th. XIII.6, p. 349

[62] Lang 1993, Th. III.1.1

[63] Choquet 1966, Lemma III.16.11

[64] Kreyszig 1999, Chapter 11

[65] Griﬃths 1995, Chapter 1

[66] Lang 1993, ch. XVII.3

[67] Lang 2002, ch. III.1, p. 121

[68] Eisenbud 1995, ch. 1.6 11.12. FOOTNOTES 131

[69] Varadarajan 1974

[70] Lang 2002, ch. XVI.7

[71] Lang 2002, ch. XVI.8

[72] Luenberger 1997, §7.13

[73] See representation theory and group representation.

[74] Lang 1993, Ch. XI.1

[75] Evans 1998, Th. 6.2.1

[76] Folland 1992, p. 349 ﬀ

[77] Gasquet & Witomski 1999, p. 150

[78] Gasquet & Witomski 1999, §4.5

[79] Gasquet & Witomski 1999, p. 57

[80] Loomis 1953, Ch. VII

[81] Ashcroft & Mermin 1976, Ch. 5

[82] Kreyszig 1988, p. 667

[83] Fourier 1822

[84] Gasquet & Witomski 1999, p. 67

[85] Ifeachor & Jervis 2002, pp. 3–4, 11

[86] Wallace Feb 1992

[87] Ifeachor & Jervis 2002, p. 132

[88] Gasquet & Witomski 1999, §10.2

[89] Ifeachor & Jervis 2002, pp. 307–310

[90] Gasquet & Witomski 1999, §10.3

[91] Schönhage & Strassen 1971

[92] Spivak 1999, ch. 3

[93] Jost 2005. See also Lorentzian manifold.

[94] Misner, Thorne & Wheeler 1973, ch. 1.8.7, p. 222 and ch. 2.13.5, p. 325

[95] Jost 2005, ch. 3.1

[96] Varadarajan 1974, ch. 4.3, Theorem 4.3.27

[97] Kreyszig 1991, §34, p. 108

[98] Eisenberg & Guy 1979

[99] Atiyah 1989

[100] Artin 1991, ch. 12

[101] Grillet, Pierre Antoine. Abstract algebra. Vol. 242. Springer Science & Business Media, 2007.

[102] Meyer 2000, Example 5.13.5, p. 436

[103] Meyer 2000, Exercise 5.13.15–17, p. 442

[104] Coxeter 1987 132 CHAPTER 11. VECTOR SPACE

11.13 References

11.13.1 Algebra

• Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1

• Blass, Andreas (1984), “Existence of bases implies the axiom of choice”, Axiomatic set theory (Boulder, Col- orado, 1983), Contemporary Mathematics 31, Providence, R.I.: American Mathematical Society, pp. 31–33, MR 763890

• Brown, William A. (1991), Matrices and vector spaces, New York: M. Dekker, ISBN 978-0-8247-8419-5

• Lang, Serge (1987), Linear algebra, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96412-6

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer- Verlag, ISBN 978-0-387-95385-4, MR 1878556

• Mac Lane, Saunders (1999), Algebra (3rd ed.), pp. 193–222, ISBN 0-8218-1646-2

• Meyer, Carl D. (2000), Matrix Analysis and Applied Linear Algebra, SIAM, ISBN 978-0-89871-454-8

• Roman, Steven (2005), Advanced Linear Algebra, Graduate Texts in Mathematics 135 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-24766-3

• Spindler, Karlheinz (1993), Abstract Algebra with Applications: Volume 1: Vector spaces and groups, CRC, ISBN 978-0-8247-9144-5

• van der Waerden, Bartel Leendert (1993), Algebra (in German) (9th ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-56799-8

11.13.2 Analysis

• Bourbaki, Nicolas (1987), Topological vector spaces, Elements of mathematics, Berlin, New York: Springer- Verlag, ISBN 978-3-540-13627-9

• Bourbaki, Nicolas (2004), Integration I, Berlin, New York: Springer-Verlag, ISBN 978-3-540-41129-1

• Braun, Martin (1993), Diﬀerential equations and their applications: an introduction to applied mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-97894-9

• BSE-3 (2001), “Tangent plane”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978- 1-55608-010-4

• Choquet, Gustave (1966), Topology, Boston, MA: Academic Press

• Dennery, Philippe; Krzywicki, Andre (1996), Mathematics for Physicists, Courier Dover Publications, ISBN 978-0-486-69193-0

• Dudley, Richard M. (1989), Real analysis and probability, The Wadsworth & Brooks/Cole Mathematics Series, Paciﬁc Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, ISBN 978-0-534-10050-6

• Dunham, William (2005), The Calculus Gallery, Princeton University Press, ISBN 978-0-691-09565-3

• Evans, Lawrence C. (1998), Partial diﬀerential equations, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0772-9

• Folland, Gerald B. (1992), Fourier Analysis and Its Applications, Brooks-Cole, ISBN 978-0-534-17094-3

• Gasquet, Claude; Witomski, Patrick (1999), Fourier Analysis and Applications: Filtering, Numerical Compu- tation, Wavelets, Texts in Applied Mathematics, New York: Springer-Verlag, ISBN 0-387-98485-2

• Ifeachor, Emmanuel C.; Jervis, Barrie W. (2001), Digital Signal Processing: A Practical Approach (2nd ed.), Harlow, Essex, England: Prentice-Hall (published 2002), ISBN 0-201-59619-9 11.13. REFERENCES 133

• Krantz, Steven G. (1999), A Panorama of Harmonic Analysis, Carus Mathematical Monographs, Washington, DC: Mathematical Association of America, ISBN 0-88385-031-1

• Kreyszig, Erwin (1988), Advanced Engineering Mathematics (6th ed.), New York: John Wiley & Sons, ISBN 0-471-85824-2

• Kreyszig, Erwin (1989), Introductory functional analysis with applications, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-50459-7, MR 992618

• Lang, Serge (1983), Real analysis, Addison-Wesley, ISBN 978-0-201-14179-5

• Lang, Serge (1993), Real and functional analysis, Berlin, New York: Springer-Verlag, ISBN 978-0-387- 94001-4

• Loomis, Lynn H. (1953), An introduction to abstract harmonic analysis, Toronto-New York–London: D. Van Nostrand Company, Inc., pp. x+190

• Schaefer, Helmut H.; Wolﬀ, M.P. (1999), Topological vector spaces (2nd ed.), Berlin, New York: Springer- Verlag, ISBN 978-0-387-98726-2

• Treves, François (1967), Topological vector spaces, distributions and kernels, Boston, MA: Academic Press

11.13.3 Historical references

• Banach, Stefan (1922), “Sur les opérations dans les ensembles abstraits et leur application aux équations inté- grales (On operations in abstract sets and their application to integral equations)" (PDF), Fundamenta Mathe- maticae (in French) 3, ISSN 0016-2736

• Bolzano, Bernard (1804), Betrachtungen über einige Gegenstände der Elementargeometrie (Considerations of some aspects of elementary geometry) (in German)

• Bourbaki, Nicolas (1969), Éléments d'histoire des mathématiques (Elements of history of mathematics) (in French), Paris: Hermann

• Dorier, Jean-Luc (1995), “A general outline of the genesis of vector space theory”, Historia Mathematica 22 (3): 227–261, doi:10.1006/hmat.1995.1024, MR 1347828

• Fourier, Jean Baptiste Joseph (1822), Théorie analytique de la chaleur (in French), Chez Firmin Didot, père et ﬁls

• Grassmann, Hermann (1844), Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik (in German), O. Wigand, reprint: Hermann Grassmann. Translated by Lloyd C. Kannenberg. (2000), Kannenberg, L.C., ed., Extension Theory, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2031-5

• Hamilton, William Rowan (1853), Lectures on Quaternions, Royal Irish Academy

• Möbius, August Ferdinand (1827), Der Barycentrische Calcul : ein neues Hülfsmittel zur analytischen Behand- lung der Geometrie (Barycentric calculus: a new utility for an analytic treatment of geometry) (in German)

• Moore, Gregory H. (1995), “The axiomatization of linear algebra: 1875–1940”, Historia Mathematica 22 (3): 262–303, doi:10.1006/hmat.1995.1025

• Peano, Giuseppe (1888), Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva (in Italian), Turin

11.13.4 Further references

• Ashcroft, Neil; Mermin, N. David (1976), Solid State Physics, Toronto: Thomson Learning, ISBN 978-0-03- 083993-1

• Atiyah, Michael Francis (1989), K-theory, Advanced Book Classics (2nd ed.), Addison-Wesley, ISBN 978-0- 201-09394-0, MR 1043170 134 CHAPTER 11. VECTOR SPACE

• Bourbaki, Nicolas (1998), Elements of Mathematics : Algebra I Chapters 1-3, Berlin, New York: Springer- Verlag, ISBN 978-3-540-64243-5 • Bourbaki, Nicolas (1989), General Topology. Chapters 1-4, Berlin, New York: Springer-Verlag, ISBN 978-3- 540-64241-1 • Coxeter, Harold Scott MacDonald (1987), Projective Geometry (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-96532-1 • Eisenberg, Murray; Guy, Robert (1979), “A proof of the hairy ball theorem”, The American Mathematical Monthly (Mathematical Association of America) 86 (7): 572–574, doi:10.2307/2320587, JSTOR 2320587 • Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics 150, Berlin, New York: Springer- Verlag, ISBN 978-0-387-94269-8, MR 1322960 • Goldrei, Derek (1996), Classic Set Theory: A guided independent study (1st ed.), London: Chapman and Hall, ISBN 0-412-60610-0 • Griffiths, David J. (1995), Introduction to Quantum Mechanics, Upper Saddle River, NJ: Prentice Hall, ISBN 0-13-124405-1 • Halmos, Paul R. (1974), Finite-dimensional vector spaces, Berlin, New York: Springer-Verlag, ISBN 978-0- 387-90093-3 • Halpern, James D. (Jun 1966), “Bases in Vector Spaces and the Axiom of Choice”, Proceedings of the Ameri- can Mathematical Society (American Mathematical Society) 17 (3): 670–673, doi:10.2307/2035388, JSTOR 2035388 • Husemoller, Dale (1994), Fibre Bundles (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387- 94087-8 • Jost, Jürgen (2005), Riemannian Geometry and Geometric Analysis (4th ed.), Berlin, New York: Springer- Verlag, ISBN 978-3-540-25907-7 • Kreyszig, Erwin (1991), Differential geometry, New York: Dover Publications, pp. xiv+352, ISBN 978-0-486- 66721-8 • Kreyszig, Erwin (1999), Advanced Engineering Mathematics (8th ed.), New York: John Wiley & Sons, ISBN 0-471-15496-2 • Luenberger, David (1997), Optimization by vector space methods, New York: John Wiley & Sons, ISBN 978- 0-471-18117-0 • Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer- Verlag, ISBN 978-0-387-98403-2 • Misner, Charles W.; Thorne, Kip; Wheeler, John Archibald (1973), Gravitation, W. H. Freeman, ISBN 978- 0-7167-0344-0 • Naber, Gregory L. (2003), The geometry of Minkowski spacetime, New York: Dover Publications, ISBN 978- 0-486-43235-9, MR 2044239 • Schönhage, A.; Strassen, Volker (1971), “Schnelle Multiplikation großer Zahlen (Fast multiplication of big numbers)" (PDF), Computing (in German) 7: 281–292, doi:10.1007/bf02242355, ISSN 0010-485X • Spivak, Michael (1999), A Comprehensive Introduction to Differential Geometry (Volume Two), Houston, TX: Publish or Perish • Stewart, Ian (1975), Galois Theory, Chapman and Hall Mathematics Series, London: Chapman and Hall, ISBN 0-412-10800-3 • Varadarajan, V. S. (1974), Lie groups, Lie algebras, and their representations, Prentice Hall, ISBN 978-0-13- 535732-3 • Wallace, G.K. (Feb 1992), “The JPEG still picture compression standard”, IEEE Transactions on Consumer Electronics 38 (1): xviii–xxxiv, doi:10.1109/30.125072, ISSN 0098-3063 • Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathe- matics 38, Cambridge University Press, ISBN 978-0-521-55987-4, OCLC 36131259, MR 1269324 11.14. EXTERNAL LINKS 135

11.14 External links

• Hazewinkel, Michiel, ed. (2001), “Vector space”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608- 010-4 • A lecture about fundamental concepts related to vector spaces (given at MIT)

• A graphical simulator for the concepts of span, linear dependency, base and dimension 136 CHAPTER 11. VECTOR SPACE

11.15 Text and image sources, contributors, and licenses

11.15.1 Text

• Augmented matrix Source: https://en.wikipedia.org/wiki/Augmented_matrix?oldid=670245141 Contributors: Giftlite, Alison, Mh, El C, 3mta3, Caleb666, Oleg Alexandrov, Bjones, StradivariusTV, GrundyCamellia, Duomillia, 48v, BiH, Octahedron80, Addshore, Shad- owdragon07, AdrianX, Mwhiz, Vanish2, Carlicus~enwiki, Sarregouset, Randomblue, Cliff, Truthnlove, Addbot, Cabwood, Luckas-bot, GrouchoBot, Nageh, Husoski, Duoduoduo, Slawekb, QEDK, ClueBot NG, Bazuz, Qetuth, YFdyh-bot, Makecat-bot and Anonymous: 24 • Coefficient Source: https://en.wikipedia.org/wiki/Coefficient?oldid=672827553 Contributors: The Anome, Rade Kutil, Heron, Michael Hardy, Silverfish, Ffransoo, Charles Matthews, SchmuckyTheCat, Bkell, Hadal, Michael Snow, Tobias Bergemann, Marc Venot, Giftlite, Bovlb, Eequor, Mormegil, Discospinster, Paul August, Rgdboer, Sam Korn, Jumbuck, Msh210, Alansohn, Gene Nygaard, Crosbie- smith, WadeSimMiser, Magister Mathematicae, MarSch, Jameshfisher, Fresheneesz, TheGreyHats, Chobot, Roboto de Ajvol, YurikBot, RobotE, Pip2andahalf, Michael Slone, GeeJo, Shreshth91, S.L.~enwiki, Nucleusboy, Mad Max, DavidHouse~enwiki, Bota47, Haemo, Spliffy, SmackBot, Maksim-e~enwiki, Skizzik, IMacWin95, Octahedron80, Sidious1701, Cybercobra, Decltype, Amtiss, Cronholm144, Mets501, Igoldste, Hynca-Hooley, Iokseng, ST47, Biblbroks, UberScienceNerd, Epbr123, Braveorca, Escarbot, AntiVandalBot, Oddity- , Res2216firestar, JAnDbot, Bongwarrior, A Hauptfleisch, Granburguesa, JaGa, Hdt83, MartinBot, Tgeairn, Pharaoh of the Wizards, Lantonov, Enuja, Jarry1250, Signalhead, VolkovBot, Someguy1221, Broadbot, Maxim, Dogah, Xanstarchild, Paolo.dL, JackSchmidt, Denisarona, ClueBot, Deviator13, Gene93k, Uncle Milty, Niceguyedc, UKoch, DragonBot, Excirial, CrazyChemGuy, Estirabot, Thingg, DumZiBoT, Hotcrocodile, Marc van Leeuwen, Stickee, Gggh, CalumH93, Addbot, Proofreader77, Atethnekos, Fgnievinski, Ronhjones, ,Zorrobot, Legobot, Luckas-bot, Yobot, SwisterTwister, Tempodivalse ,דוד שי ,Wikimichael22, Fluffernutter, AndersBot, Tide rolls Ciphers, Speller26, IRP, Piano non troppo, Darolew, Materialscientist, E2eamon, Capricorn42, Renaissancee, Mgtrevisan, 33rogers, Lu- ,Diannaa, DARTH SIDIOUS 2, EmausBot, WikitanvirBot ,علی ویکی ,cienBOT, Bkerkanator, I dream of horses, Uknighter, Vrenator RA0808, Darkfight, ZéroBot, John Cline, Chharvey, D.Lazard, ChuispastonBot, ClueBot NG, Gareth Griffith-Jones, Wcherowi, Help- ful Pixie Bot, DBigXray, Mark Arsten, Peru Serv, Gunn1t, Omulae, GoShow, Makecat-bot, Lugia2453, Frosty, Cmckain14, Neitiznot, Tango303, Hollylilholly, Graceracer525, Carlos881, MinnieBeachBum1382 and Anonymous: 196 • Coefficient matrix Source: https://en.wikipedia.org/wiki/Coefficient_matrix?oldid=546904705 Contributors: Ram einstein, Oleg Alexan- drov, SmackBot, BiT, Fetofs, Sadads, Mwhiz, Addbot, Raffamaiden, Erik9bot, Rocketchess and Anonymous: 6 • Determinant Source: https://en.wikipedia.org/wiki/Determinant?oldid=672701760 Contributors: AxelBoldt, Bryan Derksen, Tarquin, Taw, Patrick, Michael Hardy, Wshun, Delirium, Ahoerstemeier, Stevenj, Snoyes, TheEternalVortex, AugPi, Andres, Wael Ellithy, Charles Matthews, Dcoetzee, Dysprosia, Jitse Niesen, Wik, Itai, McKay, Kwantus, Rogper~enwiki, Robbot, Fredrik, Benwing, Big Jim Fae Scot- land, MathMartin, Robinh, Tobias Bergemann, Tosha, Connelly, Giftlite, Christopher Parham, Arved, Gene Ward Smith, Recentchanges, BenFrantzDale, Tegla, Lethe, Fropuff, Bkonrad, TomViza, Alison, Ssd, Rpyle731, Quadell, Anythingyouwant, Icairns, Rpchase, Jew- bacca, Asqueella, Slady, Rich Farmbrough, TedPavlic, Ardonik, ArnoldReinhold, MuDavid, Paul August, Bender235, Zaslav, Gauge, Pt, Rgdboer, Nickj, Gershwinrb, Spoon!, Reinyday, Guiltyspark, Tgr, Obradovic Goran, Nsaa, Jumbuck, Wirawan0, Msa11usec, Nachiket- vartak, Burn, Danaman5, HenkvD, Jheald, Smithereens~enwiki, Oleg Alexandrov, Alkarex, Catfive, Joriki, N3vln, Shreevatsa, LOL, StradivariusTV, Siener, ^demon, Netdragon, Graham87, Zzedar, Rjwilmsi, Ae77, R.e.b., TheIncredibleEdibleOompaLoompa, Marozols, FlaBot, Anskas, Mathbot, RexNL, Chobot, Jersey Devil, Ztutz~enwiki, Bgwhite, Gwernol, Algebraist, YurikBot, Wavelength, Personman, RussBot, Michael Slone, Josteinaj, Ino5hiro, Goffrie, Sangwine, NeilenMarais, Misza13, Crasshopper, Zwobot, Nethgirb, Entropeneur, Bota47, Jemebius, Woscafrench, Lt-wiki-bot, Cbogart2, Closedmouth, Jogers, Netrapt, Nekura, Vanka5, SmackBot, RDBury, Jagged 85, Alksub, Pedrose, Spireguy, Swerdnaneb, Shai-kun, Betacommand, Kurykh, Ian13, Oli Filth, Carbonrodney, Octahedron80, DHN- bot~enwiki, A. B., Tekhnofiend, Lhf, Trifon Triantafillidis, Adamp, Sayahoy, Kunal Bhalla, Merge, Stefano85, Balagen, Lambiam, J. Finkelstein, Dark Formal, Nat2, Cronholm144, Jim.belk, Kaarebrandt, Mets501, Cowanae, RekishiEJ, Alexandre Martins, AbsolutDan, Lavaka, Cthulhu.mythos, SkyWalker, Mrsaad31, CRGreathouse, Jackzhp, CBM, Mcstrother, Anakata, HenningThielemann, Leakeyjee, Doctormatt, Juansempere, Jrgetsin, Edwardrf, Thijs!bot, Epbr123, Headbomb, Marek69, Dmbrown00, Urdutext, Rfgeorge, AntiVandal- Bot, Salgueiro~enwiki, Rbb l181, Asmeurer, Thenub314, 01001, A-asadi, Tarif Ezaz, Coffee2theorems, Kokin, Jakob.scholbach, Sod- abottle, Pigei, Trioculite, JJ Harrison, Stdazi, Adam4445, Pensador82, Jondaman21, Heili.brenna, Cesarth~enwiki, Ulisse0, Numbo3, Rocchini, Jerry, Mikael Häggström, BrianOfRugby, Quantling, Cobi, Fylwind, Jasonevans, Cuzkatzimhut, VolkovBot, Camrn86, Jeff G., Hlevkin, Rei-bot, Anonymous Dissident, Josp-mathilde, MackSalmon, Elphion, MartinOtter, Kmhkmh, SwordSmurf, Wolfrock, Cam- puzano85, PhysPhD, Dmcq, HiDrNick, Demize, Priitliivak, Gerakibot, Icktoofay, Cwkmail, Rdmabry, Patamia, Paolo.dL, Lightmouse, JackSchmidt, AlanUS, Trashbird1240, Gejikeiji~enwiki, DonAByrd, ClueBot, Justin W Smith, The Thing That Should Not Be, Gre- walWiki, Barking Mad142, Mild Bill Hiccup, Timberframe, Sabri76, Bender2k14, EtudiantEco, Sun Creator, Kaspar.jan, EverettYou, Alexey Muranov, SchreiberBike, Thehelpfulone, Jshen6, Mcconnell3, Johnuniq, XLinkBot, Marc van Leeuwen, Nicolae Coman, San- dro.bosio, Truthnlove, Addbot, Roentgenium111, BjornPoonen, Samiswicked, Protonk, LaaknorBot, EconoPhysicist, SpBot, LinkFA- Bot, Mdnahas, Legobot, MariaDroujkova, Luckas-bot, Quadrescence, Yobot, RIBEYE special, Vanished user rt41as76lk, Mmxx, TestE- ditBot, VanishedUser sdu9aya9fasdsopa, Ipatrol, Kingpin13, Crystal whacker, Ffatossaajvazii, Materialscientist, LilHelpa, Xqbot, Dan- testyrael, J4lambert, Rayray28, Kstueve, Joejc, Logapragasan, Executive Outcomes, T8191, Spartan S58, Correnos, Lagelspeil, Rum- blethunder, Sławomir Biały, Greynose, Citation bot 1, Kd345205, SUL, AstaBOTh15, DrilBot, HRoestBot, LAncienne, Alexander Chervov, MastiBot, CarlBac, BPets, Istcol, Simon Sang, Kallikanzarid, FoxBot, Mobiusthefrost, Jordgette, Alex Sabaka, Datahaki, Cutelyaware, Unbitwise, Ijpulido, Nistra, EmausBot, Vincent Semeria, Wisapi, Bencbartlett, Alexandre Duret-Lutz, Hangitfresh, Ccan- dan, Luiscardona89, Ebrambot, Quondum, Eniagrom, Chocochipmuffin, Tolly4bolly, Gabriel10yf, Maschen, Donner60, Chewings72, JFB80, Euphrat1508, ClueBot NG, Wcherowi, Michael P. Barnett, Trivialsegfault, Chester Markel, Movses-bot, Snotbot, TreyGreer62, Khabgood, Helpful Pixie Bot, Supreme fascist, Cjkstephenson, Vibhor1997, Myshka spasayet lva, Cmmthf, René Vápeník, Thiagowfx, Dexbot, Deltahedron, Mkhan3189, Sebhofer, Joeinwiki, Trompedo, Colombomattia89, E290341, Babitaarora, Westwood0.137, Improb- able keeler, Hkmcszz, Abitslow, Leegrc, GeoffreyT2000, KH-1, Loraof, Premshah95, Sunnsambhi, Rywais, Ajbaumann and Anonymous: 390 • Greatest common divisor Source: https://en.wikipedia.org/wiki/Greatest_common_divisor?oldid=671260152 Contributors: AxelBoldt, Carey Evans, Bryan Derksen, Zundark, Tarquin, Taw, XJaM, Hannes Hirzel, Michael Hardy, Palnatoke, SGBailey, TakuyaMurata, Poor Yorick, Nikai, Ideyal, Revolver, Charles Matthews, Dcoetzee, Bemoeial, Dysprosia, Jitse Niesen, Hyacinth, SirJective, JorgeGG, Fredrik, Henrygb, Jleedev, Tosha, Giftlite, Tom harrison, Herbee, Jason Quinn, Nayuki, Espetkov, Blankfaze, Azuredu, Karl Dickman, Mormegil, Guanabot, KneeLess, Paul August, Bender235, ESkog, EmilJ, Spoon!, Obradovic Goran, Haham hanuka, LutzL, Jumbuck, Silver hr, Arthena, Kenyon, Oleg Alexandrov, Linas, LOL, Ruud Koot, WadeSimMiser, Qwertyus, Josh Parris, Rjwilmsi, Staecker, Bryan H Bell, 11.15. 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VECTOR SPACE

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Frost, Tim32, Alexbot, Leonard^Bloom, Kaspar.jan, Newyorxico, Thingg, XLinkBot, SilvonenBot, JinJian, Addbot, SahilK7654, Steve.jaramillov~enwiki, Fgnievinski, IgorCarron, II Mus- LiM HyBRiD II, Calle, CinchBug, KamikazeBot, AnomieBOT, Jim1138, Vanakaris, Twri, ArthurBot, Xqbot, Matttoothman, ToLLIa, EmausBot, John of ,בן גרשון ,Tyrol5, Frankie0607, Universalss, Constructive editor, Tavernsenses, BenzolBot, Jauhienij, Duoduoduo Reading, Inframaut, DotKuro, Tommy2010, Slawekb, D.Lazard, OnePt618, Donner60, ChuispastonBot, Anita5192, ClueBot NG, Ooz dot ie, Movses-bot, 3rdiw, Helpful Pixie Bot, Guy vandegrift, Hz.tiang, René Vápeník, BattyBot, Pratyush Sarkar, Stigmatella aurantiaca, Boesball, Mark L MacDonald, Webclient101, Frosty, Graphium, Makalrfekt, Pdecalculus, Lakun.patra, Loraof and Anonymous: 187 • Vector space Source: https://en.wikipedia.org/wiki/Vector_space?oldid=672138139 Contributors: AxelBoldt, Bryan Derksen, Zundark, The Anome, Taw, Awaterl, Youandme, N8chz, Olivier, Tomo, Patrick, Michael Hardy, Tim Starling, Wshun, Nixdorf, Kku, Gabbe, Wapcaplet, TakuyaMurata, Pcb21, Iulianu, Glenn, Ciphergoth, Dysprosia, Jitse Niesen, Jogloran, Phys, Kwantus, Aenar, Robbot, Ro- manm, P0lyglut, Tobias Bergemann, Giftlite, BenFrantzDale, Lethe, MathKnight, Fropuff, Waltpohl, Andris, Daniel Brockman, Python eggs, Chowbok, Sreyan, Lockeownzj00, MarkSweep, Profvk, Maximaximax, Barnaby dawson, Mh, Klaas van Aarsen, TedPavlic, Rama, Smyth, Notinasnaid, Paul August, Bender235, Rgdboer, Shoujun, Army1987, Cmdrjameson, Stephen Bain, Tsirel, Msh210, Orimosen- zon, ChrisUK, Ncik~enwiki, Eric Kvaalen, ABCD, Sligocki, Jheald, Eddie Dealtry, Dirac1933, Woodstone, Kbolino, Oleg Alexandrov, Woohookitty, Mindmatrix, ^demon, Hfarmer, Mpatel, MFH, Graham87, Ilya, Rjwilmsi, Koavf, MarSch, Omnieiunium, Salix alba, Titoxd, FlaBot, VKokielov, Therearenospoons, Nihiltres, Ssafarik, Srleffler, Kri, R160K, Chobot, Gwernol, Algebraist, YurikBot, Wave- length, Spacepotato, Hairy Dude, RussBot, Michael Slone, CambridgeBayWeather, Rick Norwood, Kinser, Guruparan, Trovatore, Van- ished user 1029384756, Nick, Bota47, BraneJ, Martinwilke1980, Antiduh, Lonerville, Netrapt, Curpsbot-unicodify, Cjfsyntropy, Paul D. 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