DOCSLIB.ORG

1 Special Types of SQUARE Matrices

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1 Special Types of SQUARE Matrices 1 Special Types of SQUARE Matrices Name Description Examples 2 3 Diagonal a = 0 for any i 6= j 4 0 0 0 0 ij 2 3 3 0 0 60 0 0 0 07 6 7 Notation: That is, all entries 40 1 05 or 60 0 11 0 07 6 7 Diag(a11; :::; ann) off the main diagonal 0 0 7 40 0 0 6 05 must be 0. Diago- 0 0 0 0 2 nal entries can be These matrices can be denoted by Diag(3; 1; 7) anything, including 0. and Diag(4; 0; 11; 6; 2), respectively. Identity (I ) A diagonal matrix n 21 0 03 whose main-diagonal 1 0 I = 0 1 0 or I = entries are all 1. 3 4 5 2 0 1 0 0 1 Scalar A diagonal matrix of 25 0 0 03 the form r(In) for 60 5 0 07 −3 0 5(I4) = 6 7 or −3(I2) = some scalar r. 40 0 5 05 0 −3 0 0 0 5 Upper Triangular A = [aij] with aij = 0 for all i > j 25 4 8 23 That is, entries 21 2 43 0 2 1 3 \below" the main 0 3 1 or 6 7 4 5 60 0 0 17 diagonal must be 0. 0 0 2 4 5 0 0 0 5 Entries on or \above" the main diagonal can be anything, including 0. Lower Triangular A = [aij] with aij = 0 for all i < j 24 0 0 0 03 2 3 That is, entries 9 0 0 61 3 0 0 07 6 7 \above" the main 44 2 05 or 60 4 5 0 07 6 7 diagonal must be 0. 1 3 8 41 3 5 9 05 Entries on or \below" 5 6 2 9 4 the main diagonal can be anything, including 0. Symmetric AT = A 21 2 3 4 5 3 2 3 4 12 3 62 0 7 8 9 7 (only applies or 6 7 to matrices with 412 1 85 or 63 7 2 6 107 3 8 5 64 8 6 3 1 7 real entries) aij = aji 4 5 5 9 10 1 7 Skew Symmetric AT = −A 2 0 7 6 −13 2 0 1 −33 (only applies or −7 0 0 2 −1 0 8 or 6 7 to matrices with 4 5 6−6 0 0 −47 3 8 0 4 5 real entries) aij = −aji 1 −2 4 0 NOTE: In the definitions of symmetric and skew symmetric matrices, we require that all entries be real numbers. For the other types of matrices mentioned, entires could (technically) be complex numbers, but we won't work with complex entries until later in the semester. Moreover, entries do not all have to be integers, even though the examples include only integer entries. This was merely for convenience and clarity. Question: Notice that in the examples I gave of skew symmetric matrices, the entries on the main diagonal were all 0. Does this always happen for skew symmetric matrices? If so, why? Question: What types of matrices can be both upper and lower triangular? Question: I claim that for any n, there is only one n × n square matrix which is both symmetric and skew symmetric. Am I right? If so, what is the unique matrix with this property? Question: Consider an n × n diagonal matrix A = Diag(a11; a22; :::; ann). What happens when you take powers of A? Compute A2 and A3. Can you make a guess at a formula for Ap for any positive integer p? Fact: If A is an m × n matrix, then ImA = A and AIn = A. Fact: (This is Exercise 1.5.29 in the book.) If A is an n × n square matrix, then we can show that A = S + K, where S is symmetric and K is skew symmetric. That is, A can be decomposed as the sum of a skew matrix and a skew symmetric matrix. Also, this decomposition is unique. 2 Powers of Matrices • A2 should mean AA. If A is size m × n, then AA = A2 only makes sense if n = m. In other words, we can only take powers of square matrices. • If A is a square matrix of size n × n and if p is a positive integer, then Ap = AA ··· A | {z } p times 0 • By convention, if A is size n × n, we say that A = In. • FACT: If p and q are nonnegative integers and A is a square matrix, then ApAq = Ap+q and (Ap)q = Apq • CAUTION: (AB)p and ApBp may not be (and usually are not) the same. (See Exercise 1.5.9) 3 Partitioned Matrices and Block Multiplication Please read about these topics in Section 1.5 of the textbook. It starts at the bottom of Pg. 43 and ends in the middle of Pg. 46. Then try out block multiplication by doing Exercise 1.5.56. Block multiplication helps computers complete matrix multiplication more quickly. However, when you are doing matrix multiplication by hand, block multiplication doesn't have major advantages..
Load more

Recommended publications
  • 21. Orthonormal Bases
    21. Orthonormal Bases The canonical/standard basis 011 001 001 B C B C B C B0C B1C B0C e1 = B.C ; e2 = B.C ; : : : ; en = B.C B.C B.C B.C @.A @.A @.A 0 0 1 has many useful properties. • Each of the standard basis vectors has unit length: q p T jjeijj = ei ei = ei ei = 1: • The standard basis vectors are orthogonal (in other words, at right angles or perpendicular). T ei ej = ei ej = 0 when i 6= j This is summarized by ( 1 i = j eT e = δ = ; i j ij 0 i 6= j where δij is the Kronecker delta. Notice that the Kronecker delta gives the entries of the identity matrix. Given column vectors v and w, we have seen that the dot product v w is the same as the matrix multiplication vT w. This is the inner product on n T R . We can also form the outer product vw , which gives a square matrix. 1 The outer product on the standard basis vectors is interesting. Set T Π1 = e1e1 011 B C B0C = B.C 1 0 ::: 0 B.C @.A 0 01 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 0 . T Πn = enen 001 B C B0C = B.C 0 0 ::: 1 B.C @.A 1 00 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 1 In short, Πi is the diagonal square matrix with a 1 in the ith diagonal position and zeros everywhere else.
    [Show full text]
  • Partitioned (Or Block) Matrices This Version: 29 Nov 2018
    Partitioned (or Block) Matrices This version: 29 Nov 2018 Intermediate Econometrics / Forecasting Class Notes Instructor: Anthony Tay It is frequently convenient to partition matrices into smaller sub-matrices. e.g. 2 3 2 1 3 2 3 2 1 3 4 1 1 0 7 4 1 1 0 7 A B (2×2) (2×3) 3 1 1 0 0 = 3 1 1 0 0 = C I 1 3 0 1 0 1 3 0 1 0 (3×2) (3×3) 2 0 0 0 1 2 0 0 0 1 The same matrix can be partitioned in several different ways. For instance, we can write the previous matrix as 2 3 2 1 3 2 3 2 1 3 4 1 1 0 7 4 1 1 0 7 a b0 (1×1) (1×4) 3 1 1 0 0 = 3 1 1 0 0 = c D 1 3 0 1 0 1 3 0 1 0 (4×1) (4×4) 2 0 0 0 1 2 0 0 0 1 One reason partitioning is useful is that we can do matrix addition and multiplication with blocks, as though the blocks are elements, as long as the blocks are conformable for the operations. For instance: A B D E A + D B + E (2×2) (2×3) (2×2) (2×3) (2×2) (2×3) + = C I C F 2C I + F (3×2) (3×3) (3×2) (3×3) (3×2) (3×3) A B d E Ad + BF AE + BG (2×2) (2×3) (2×1) (2×3) (2×1) (2×3) = C I F G Cd + F CE + G (3×2) (3×3) (3×1) (3×3) (3×1) (3×3) | {z } | {z } | {z } (5×5) (5×4) (5×4) 1 Intermediate Econometrics / Forecasting 2 Examples (1) Let 1 2 1 1 2 1 c 1 4 2 3 4 2 3 h i A = = = a a a and c = c 1 2 3 2 3 0 1 3 0 1 c 0 1 3 0 1 3 3 c1 h i then Ac = a1 a2 a3 c2 = c1a1 + c2a2 + c3a3 c3 The product Ac produces a linear combination of the columns of A.
    [Show full text]
  • Handout 9 More Matrix Properties; the Transpose
    Handout 9 More matrix properties; the transpose Square matrix properties These properties only apply to a square matrix, i.e. n £ n. ² The leading diagonal is the diagonal line consisting of the entries a11, a22, a33, . ann. ² A diagonal matrix has zeros everywhere except the leading diagonal. ² The identity matrix I has zeros o® the leading diagonal, and 1 for each entry on the diagonal. It is a special case of a diagonal matrix, and A I = I A = A for any n £ n matrix A. ² An upper triangular matrix has all its non-zero entries on or above the leading diagonal. ² A lower triangular matrix has all its non-zero entries on or below the leading diagonal. ² A symmetric matrix has the same entries below and above the diagonal: aij = aji for any values of i and j between 1 and n. ² An antisymmetric or skew-symmetric matrix has the opposite entries below and above the diagonal: aij = ¡aji for any values of i and j between 1 and n. This automatically means the digaonal entries must all be zero. Transpose To transpose a matrix, we reect it across the line given by the leading diagonal a11, a22 etc. In general the result is a di®erent shape to the original matrix: a11 a21 a11 a12 a13 > > A = A = 0 a12 a22 1 [A ]ij = A : µ a21 a22 a23 ¶ ji a13 a23 @ A > ² If A is m £ n then A is n £ m. > ² The transpose of a symmetric matrix is itself: A = A (recalling that only square matrices can be symmetric).
    [Show full text]
  • Week 8-9. Inner Product Spaces. (Revised Version) Section 3.1 Dot Product As an Inner Product
    Math 2051 W2008 Margo Kondratieva Week 8-9. Inner product spaces. (revised version) Section 3.1 Dot product as an inner product. Consider a linear (vector) space V . (Let us restrict ourselves to only real spaces that is we will not deal with complex numbers and vectors.) De¯nition 1. An inner product on V is a function which assigns a real number, denoted by < ~u;~v> to every pair of vectors ~u;~v 2 V such that (1) < ~u;~v>=< ~v; ~u> for all ~u;~v 2 V ; (2) < ~u + ~v; ~w>=< ~u;~w> + < ~v; ~w> for all ~u;~v; ~w 2 V ; (3) < k~u;~v>= k < ~u;~v> for any k 2 R and ~u;~v 2 V . (4) < ~v;~v>¸ 0 for all ~v 2 V , and < ~v;~v>= 0 only for ~v = ~0. De¯nition 2. Inner product space is a vector space equipped with an inner product. Pn It is straightforward to check that the dot product introduces by ~u ¢ ~v = j=1 ujvj is an inner product. You are advised to verify all the properties listed in the de¯nition, as an exercise. The dot product is also called Euclidian inner product. De¯nition 3. Euclidian vector space is Rn equipped with Euclidian inner product < ~u;~v>= ~u¢~v. De¯nition 4. A square matrix A is called positive de¯nite if ~vT A~v> 0 for any vector ~v 6= ~0. · ¸ 2 0 Problem 1. Show that is positive de¯nite. 0 3 Solution: Take ~v = (x; y)T . Then ~vT A~v = 2x2 + 3y2 > 0 for (x; y) 6= (0; 0).
    [Show full text]
  • New Foundations for Geometric Algebra1
    Text published in the electronic journal Clifford Analysis, Clifford Algebras and their Applications vol. 2, No. 3 (2013) pp. 193-211 New foundations for geometric algebra1 Ramon González Calvet Institut Pere Calders, Campus Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès, Spain E-mail : [email protected] Abstract. New foundations for geometric algebra are proposed based upon the existing isomorphisms between geometric and matrix algebras. Each geometric algebra always has a faithful real matrix representation with a periodicity of 8. On the other hand, each matrix algebra is always embedded in a geometric algebra of a convenient dimension. The geometric product is also isomorphic to the matrix product, and many vector transformations such as rotations, axial symmetries and Lorentz transformations can be written in a form isomorphic to a similarity transformation of matrices. We collect the idea Dirac applied to develop the relativistic electron equation when he took a basis of matrices for the geometric algebra instead of a basis of geometric vectors. Of course, this way of understanding the geometric algebra requires new definitions: the geometric vector space is defined as the algebraic subspace that generates the rest of the matrix algebra by addition and multiplication; isometries are simply defined as the similarity transformations of matrices as shown above, and finally the norm of any element of the geometric algebra is defined as the nth root of the determinant of its representative matrix of order n. The main idea of this proposal is an arithmetic point of view consisting of reversing the roles of matrix and geometric algebras in the sense that geometric algebra is a way of accessing, working and understanding the most fundamental conception of matrix algebra as the algebra of transformations of multiple quantities.
    [Show full text]
  • Matrix Determinants
    MATRIX DETERMINANTS Summary Uses ................................................................................................................................................. 1 1‐ Reminder ‐ Definition and components of a matrix ................................................................ 1 2‐ The matrix determinant .......................................................................................................... 2 3‐ Calculation of the determinant for a matrix ................................................................. 2 4‐ Exercise .................................................................................................................................... 3 5‐ Definition of a minor ............................................................................................................... 3 6‐ Definition of a cofactor ............................................................................................................ 4 7‐ Cofactor expansion – a method to calculate the determinant ............................................... 4 8‐ Calculate the determinant for a matrix ........................................................................ 5 9‐ Alternative method to calculate determinants ....................................................................... 6 10‐ Exercise .................................................................................................................................... 7 11‐ Determinants of square matrices of dimensions 4x4 and greater ........................................
    [Show full text]
  • 6 Inner Product Spaces
    Lectures 16,17,18 6 Inner Product Spaces 6.1 Basic Definition Parallelogram law, the ability to measure angle between two vectors and in particular, the concept of perpendicularity make the euclidean space quite a special type of a vector space. Essentially all these are consequences of the dot product. Thus, it makes sense to look for operations which share the basic properties of the dot product. In this section we shall briefly discuss this. Definition 6.1 Let V be a vector space. By an inner product on V we mean a binary operation, which associates a scalar say u, v for each pair of vectors u, v) in V, satisfying h i the following properties for all u, v, w in V and α, β any scalar. (Let “ ” denote the complex − conjugate of a complex number.) (1) u, v = v, u (Hermitian property or conjugate symmetry); h i h i (2) u, αv + βw = α u, v + β u, w (sesquilinearity); h i h i h i (3) v, v > 0 if v =0 (positivity). h i 6 A vector space with an inner product is called an inner product space. Remark 6.1 (i) Observe that we have not mentioned whether V is a real vector space or a complex vector space. The above definition includes both the cases. The only difference is that if K = R then the conjugation is just the identity. Thus for real vector spaces, (1) will becomes ‘symmetric property’ since for a real number c, we have c¯ = c. (ii) Combining (1) and (2) we obtain (2’) αu + βv, w =α ¯ u, w + β¯ v, w .
    [Show full text]
  • 9. Properties of Matrices Block Matrices
    9. Properties of Matrices Block Matrices It is often convenient to partition a matrix M into smaller matrices called blocks, like so: 01 2 3 11 ! B C B4 5 6 0C A B M = B C = @7 8 9 1A C D 0 1 2 0 01 2 31 011 B C B C Here A = @4 5 6A, B = @0A, C = 0 1 2 , D = (0). 7 8 9 1 • The blocks of a block matrix must fit together to form a rectangle. So ! ! B A C B makes sense, but does not. D C D A • There are many ways to cut up an n × n matrix into blocks. Often context or the entries of the matrix will suggest a useful way to divide the matrix into blocks. For example, if there are large blocks of zeros in a matrix, or blocks that look like an identity matrix, it can be useful to partition the matrix accordingly. • Matrix operations on block matrices can be carried out by treating the blocks as matrix entries. In the example above, ! ! A B A B M 2 = C D C D ! A2 + BC AB + BD = CA + DC CB + D2 1 Computing the individual blocks, we get: 0 30 37 44 1 2 B C A + BC = @ 66 81 96 A 102 127 152 0 4 1 B C AB + BD = @10A 16 0181 B C CA + DC = @21A 24 CB + D2 = (2) Assembling these pieces into a block matrix gives: 0 30 37 44 4 1 B C B 66 81 96 10C B C @102 127 152 16A 4 10 16 2 This is exactly M 2.
    [Show full text]
  • Notes on Change of Bases Northwestern University, Summer 2014
    Notes on Change of Bases Northwestern University, Summer 2014 Let V be a finite-dimensional vector space over a field F, and let T be a linear operator on V . Given a basis (v1; : : : ; vn) of V , we've seen how we can define a matrix which encodes all the information about T as follows. For each i, we can write T vi = a1iv1 + ··· + anivn 2 for a unique choice of scalars a1i; : : : ; ani 2 F. In total, we then have n scalars aij which we put into an n × n matrix called the matrix of T relative to (v1; : : : ; vn): 0 1 a11 ··· a1n B . .. C M(T )v := @ . A 2 Mn;n(F): an1 ··· ann In the notation M(T )v, the v showing up in the subscript emphasizes that we're taking this matrix relative to the specific bases consisting of v's. Given any vector u 2 V , we can also write u = b1v1 + ··· + bnvn for a unique choice of scalars b1; : : : ; bn 2 F, and we define the coordinate vector of u relative to (v1; : : : ; vn) as 0 1 b1 B . C n M(u)v := @ . A 2 F : bn In particular, the columns of M(T )v are the coordinates vectors of the T vi. Then the point of the matrix M(T )v is that the coordinate vector of T u is given by M(T u)v = M(T )vM(u)v; so that from the matrix of T and the coordinate vectors of elements of V , we can in fact reconstruct T itself.
    [Show full text]
  • Properties of Transpose
    3.2, 3.3 Inverting Matrices P. Danziger Properties of Transpose Transpose has higher precedence than multiplica- tion and addition, so T T T T AB = A B and A + B = A + B As opposed to the bracketed expressions (AB)T and (A + B)T Example 1 1 2 1 1 0 1 Let A = ! and B = !. 2 5 2 1 1 0 Find ABT , and (AB)T . T 1 1 1 2 1 1 0 1 1 2 1 0 1 ABT = ! ! = ! 0 1 2 5 2 1 1 0 2 5 2 B C @ 1 0 A 2 3 = ! 4 7 Whereas (AB)T is undefined. 1 3.2, 3.3 Inverting Matrices P. Danziger Theorem 2 (Properties of Transpose) Given ma- trices A and B so that the operations can be pre- formed 1. (AT )T = A 2. (A + B)T = AT + BT and (A B)T = AT BT − − 3. (kA)T = kAT 4. (AB)T = BT AT 2 3.2, 3.3 Inverting Matrices P. Danziger Matrix Algebra Theorem 3 (Algebraic Properties of Matrix Multiplication) 1. (k + `)A = kA + `A (Distributivity of scalar multiplication I) 2. k(A + B) = kA + kB (Distributivity of scalar multiplication II) 3. A(B + C) = AB + AC (Distributivity of matrix multiplication) 4. A(BC) = (AB)C (Associativity of matrix mul- tiplication) 5. A + B = B + A (Commutativity of matrix ad- dition) 6. (A + B) + C = A + (B + C) (Associativity of matrix addition) 7. k(AB) = A(kB) (Commutativity of Scalar Mul- tiplication) 3 3.2, 3.3 Inverting Matrices P. Danziger The matrix 0 is the identity of matrix addition.
    [Show full text]
  • Block Matrices in Linear Algebra
    PRIMUS Problems, Resources, and Issues in Mathematics Undergraduate Studies ISSN: 1051-1970 (Print) 1935-4053 (Online) Journal homepage: https://www.tandfonline.com/loi/upri20 Block Matrices in Linear Algebra Stephan Ramon Garcia & Roger A. Horn To cite this article: Stephan Ramon Garcia & Roger A. Horn (2020) Block Matrices in Linear Algebra, PRIMUS, 30:3, 285-306, DOI: 10.1080/10511970.2019.1567214 To link to this article: https://doi.org/10.1080/10511970.2019.1567214 Accepted author version posted online: 05 Feb 2019. Published online: 13 May 2019. Submit your article to this journal Article views: 86 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=upri20 PRIMUS, 30(3): 285–306, 2020 Copyright # Taylor & Francis Group, LLC ISSN: 1051-1970 print / 1935-4053 online DOI: 10.1080/10511970.2019.1567214 Block Matrices in Linear Algebra Stephan Ramon Garcia and Roger A. Horn Abstract: Linear algebra is best done with block matrices. As evidence in sup- port of this thesis, we present numerous examples suitable for classroom presentation. Keywords: Matrix, matrix multiplication, block matrix, Kronecker product, rank, eigenvalues 1. INTRODUCTION This paper is addressed to instructors of a first course in linear algebra, who need not be specialists in the field. We aim to convince the reader that linear algebra is best done with block matrices. In particular, flexible thinking about the process of matrix multiplication can reveal concise proofs of important theorems and expose new results. Viewing linear algebra from a block-matrix perspective gives an instructor access to use- ful techniques, exercises, and examples.
    [Show full text]
  • Arxiv:1711.11297V3 [Math.RA]
    ON LOCAL AUTOMORPHISMS OF sl2 T. BECKER, J. ESCOBAR SALSEDO, C. SALAS, AND R. TURDIBAEV Abstract. We establish that the set of local automorphisms LAut(sl2) is the ± group Aut (sl2) of all automorphisms and anti-automorphisms. For n ≥ 3 we prove that anti-automorphisms are local automorphisms of sln. Introduction The idea of a local automorphism dates back to 1988 [1], where for a given set of mappings S from a set X into a set Y , Larson suggests to call a mapping θ to be interpolating S if for each x ∈ X there is an element Sx ∈ S, depending on x, with θ(x) = Sx(x). Set X = Y = A to be an algebra over a field F and S = Aut(A) the set of automorphisms of A, then a linear endomorphism ∆ of A is called a local automorphism if it interpolates the set of automorphisms of A. Local automorphisms have been introduced and studied for the algebra B(X) of all bounded linear operators on infinite dimensional Banach space X by Larson and Sourour [2]. It has been proven for the associative algebra of square matrices Mn(C) that any local automorphism is either an automorphism or an anti-automorphism [2]. Investigation of local automorphisms continues for some subalgebras of Mn(C). In [3] a description of the local automorphisms of a finite-dimensional CSL algebra is given. A CSL algebra, or a digraph algebra, is an algebra which is spanned by a set of matrices which contains all diagonal matrix units {Eii}1≤i≤n and is closed under multiplication.
    [Show full text]