DOCSLIB.ORG

Complex Inner Product Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Complex Inner Product Spaces MATH 355 Supplemental Notes Complex Inner Product Spaces Complex Inner Product Spaces The Cn spaces The prototypical (and most important) real vector spaces are the Euclidean spaces Rn. Any study of complex vector spaces will similar begin with Cn. As a set, Cn contains vectors of length n whose entries are complex numbers. Thus, 2 i ` 3 5i C3, » ´ fi P i — ffi – fl 5, 1 is an element found both in R2 and C2 (and, indeed, all of Rn is found in Cn), and 0, 0, 0, 0 p ´ q p q serves as the zero element in C4. Addition and scalar multiplication in Cn is done in the analogous way to how they are performed in Rn, except that now the scalars are allowed to be nonreal numbers. Thus, to rescale the vector 3 i, 2 3i by 1 3i, we have p ` ´ ´ q ´ 3 i 1 3i 3 i 6 8i 1 3i ` p ´ qp ` q ´ . p ´ q « 2 3iff “ « 1 3i 2 3i ff “ « 11 3iff ´ ´ p ´ qp´ ´ q ´ ` Given the notation 3 2i for the complex conjugate 3 2i of 3 2i, we adopt a similar notation ` ´ ` when we want to take the complex conjugate simultaneously of all entries in a vector. Thus, 3 4i 3 4i ´ ` » 2i fi » 2i fi if z , then z ´ . “ “ — 2 5iffi — 2 5iffi —´ ` ffi —´ ´ ffi — 1 ffi — 1 ffi — ´ ffi — ´ ffi – fl – fl Both z and z are vectors in C4. In general, if the entries of z are all real numbers, then z z. “ The inner product in Cn In Rn, the length of a vector x ?x x is a real, nonnegative number. The modulus, or length, } }“ ¨ of a complex number z a ib is real and nonnegative as well: “ ` z zz a ib a ib a2 b2, or z 2 zz. | |“ “ p ` qp ´ q“ ` | | “ a b a A natural idea, therefore, is to define an inner product between vectors z z ,...,z , w “p1 nq “ w ,...,w Cn in this manner: p 1 nqP n z, w z w w z w z wHz zTw. (1) x y “ j j “ 1 1 `¨¨¨` n n “ “ j 1 ÿ“ 2 MATH 355 Supplemental Notes Complex Inner Product Spaces Here, wH stands for wT, the conjugate transpose of w. For instance, 3 4i ´ » 2i fi if z then zH zT 3 4i 2i 2 5i 1 . “ 2 5 “ “ ` ´ ´ ´ ´ 1 4 — iffi ˆ —´ ` ffi ” ı — 1 ffi — ´ ffi4 1 – fl ˆ Remarks On p. 439 (Section 9.2) of Strang’s text, he defines the inner product of complex vectors u, • v to be the conjugate transpose of the first vector multiplied by the second—i.e., uHv.A few paragraphs down, he acknowledges that some authors do it as we have, the conjugate transpose of the second vector times the first one. The two definitions do not yield the same result. For example, if u 2 i, 1 3i, 8 and v i, 3 2i, 1 i , then my definition of “p ` ´ q “p´ ` ´ q inner product between u and v gives 2 i ` u, v vHu i 3 2i 1 i 1 3i 4 i, x y “ “ ´ ` » ´ fi “ ´ ” ı 8 — ffi – fl but Strang’s definition, which would coincide with my v, u , yields x y i ´ uHv 2 i 1 3i 8 3 2i 4 i. “ ´ ` » ` fi “ ` ” ı 1 i — ´ ffi – fl That is, one is always the complex conjugate of the other. Though my sample is limited, the linear algebra books with which I am familiar define the inner product in Cn as I have above, not the other (Strang’s) way. He is correct that it is a matter of preference which definition you use. The inner product of vectors in Cn no longer exclusively produces real numbers, as seen in • the example above. However, when taking an inner product of z z , z ,...,z with itself, “p 1 2 nq the result n n z, z zHz z z z 2, x y “ “ j j “ | j| j 1 j 1 ÿ“ ÿ“ is the sum of the moduli of the components of z, guaranteed to be nonnegative. Thus, we define length for vectors z in Cn to be z z, z , } }“ x y b and note that the only instance in which z 0 is when z is, itself, the zero vector. } }“ When the entries of z, w are all real numbers (that is, z, w Rn), our definition for inner • P product exactly matches the dot product—that is, z, w z w. x y “ ¨ 3 MATH 355 Supplemental Notes Complex Inner Product Spaces The essential list of properties that the inner product in Cn has, for all vectors u, v, w Cn and all P scalars c, is (i) v, v 0. x y • (ii) v, v 0 implies v 0. x y “ “ (iii) u w, v u, v w, v . x ` y “ x y ` x y (iv) cu, v c u, v . x y “ x y (v) u, v v, u . x y “ x y Note that (iii) and (v) together imply that u, v w u, v u, w , x ` y “ x y ` x y while (iv) and (v) together give that u, cv c u, v . x y “ x y Conjugate transpose of a matrix Suppose, now, that A is an m-by-n matrix whose entries are complex numbers. The idea of a conjugate transpose AH makes sense, as it did for vectors in Cn. In fact, it is precisely what is computed when the prime symbol is invoked in Octave. octave:127> A = randi(7,3,2) 4 ones(3,2) + (randi (5,3,2) 4 ones(3,2)) i ⌥ ´ ⇤ ´ ⇤ ⇤ ⌅ A = 2 3i 2 + 1i ´ ´ 0 1i 1 1i ´ ´ ´ 2 2i 1 + 1i ´ octave:128> A’ ans = 2 + 3i 0 + 1i 2 + 2i ´ 2 1i 1 + 1i 1 1i ´ ´ ´ ⌃ ⇧ Of course, taking the conjugate transpose of a matrix twice returns one to the original: AH H A. p q “ For complex matrices A, B of appropriate size, one can take the conjugate transpose of the product AB. If we denote the matrix full of conjugates of entries found in A by A, then we have T T AB H AB T A B T B A BHAH. p q “p q “p q “ “ 4 MATH 355 Supplemental Notes Symmetric Matrices In particular, the conjugate transpose of a matrix-vector product Av is Av H vHAH, p q “ and if we need to take an inner product between Au and v, we have the convenient formula Au, v vHAu AHv Hu u, AHv . (2) x y “ “p q “ @ D Formula (2) holds whether or not A is square. In the context of matrices, AH is sometimes called the adjoint matrix (as opposed to the conjugate transpose). If the entries in A are real numbers only, then AH AT. Any square matrix which “ satisfies AH A is said to be self-adjoint, or Hermitian. “ Symmetric Matrices We use the term symmetric to describe a matrix A whose entries are real numbers, and which satisfies AT A. Note that this implies a symmetric matrix is self-adjoint. Symmetric matrices are “ very important in applications. They have some very favorable properties. The first is that their eigenvalues are real. Theorem 1: Eigenvalues of a symmetric matrix are real. Proof: To prove this, we note first that any complex number z can be expressed in the form z a ib; here a, b are real numbers, called the real and imaginary parts of “ ` z, respectively. The number z is, in fact, real precisely when its imaginary part b 0. “ Furthermore, the di↵erence of z and its conjugate is z z a ib a ib i 2b , ´ “p` q´p ´ q“ p q which is zero if and only if z R. P Now, suppose λ, v is an eigenpair (with v , 0) of a symmetric matrix A. Consider p q the quantity λ v 2, which may alternatively be expressed as } } λ v, v λv, v Av, v v, AHv v, Av v,λv λ v, v . x y “ x y “ x y “ “ x y “ x y “ x y @ D 5 MATH 355 Supplemental Notes Symmetric Matrices Subtracting the expression at one end from that on the other gives 0 λ v, v λ v, v λ λ v, v λ λ v 2. “ x y ´ x y “p ´ q x y “p ´ q} } Since v , 0, it follows that λ λ 0, which implies λ R. ⇤ } } ´ “ P As well, symmetric matrices generate eigenvectors which are naturally orthogonal. Theorem 2: Eigenvectors corresponding to distinct eigenvalues of a symmetric matrix are orthogonal. Proof: Suppose µ, u , λ, v are both eigenpairs of a symmetric matrix A with µ , λ. p q p q By the previous theorem, µ and λ are real numbers, so λ λ. We have “ µ λ u, v µ u, v λ u, v µu, v u,λv Au, v u, Av p ´ q x y “ x y ´ x y “ x y ´ x y “ x y ´ x y u, AHv u, Av u, Av u, Av 0. “ ´ x y “ x y ´ x y “ @ D Since µ λ u, v 0 and µ λ , 0, it follows that u, v 0. ⇤ p ´ q x y “ ´ x y “ Now, if A is symmetric and has n distinct eigenvalues (all real, of course) λ1,...,λn, then the corresponding eigenspaces Null A λ I are all 1-dimensional (since GM = AM = 1 for • p ´ j q each eigenvalue) with basis vector vj, eigenvectors are the vectors v ,...,v are linearly independent (proved earlier), and form a basis of Rn, and • t 1 nu the basis v ,...,v is orthogonal, not requiring a Gram-Schmidt process to make them so.
Load more

Recommended publications
  • Math 651 Homework 1 - Algebras and Groups Due 2/22/2013
    Math 651 Homework 1 - Algebras and Groups Due 2/22/2013 1) Consider the Lie Group SU(2), the group of 2 × 2 complex matrices A T with A A = I and det(A) = 1. The underlying set is z −w jzj2 + jwj2 = 1 (1) w z with the standard S3 topology. The usual basis for su(2) is 0 i 0 −1 i 0 X = Y = Z = (2) i 0 1 0 0 −i (which are each i times the Pauli matrices: X = iσx, etc.). a) Show this is the algebra of purely imaginary quaternions, under the commutator bracket. b) Extend X to a left-invariant field and Y to a right-invariant field, and show by computation that the Lie bracket between them is zero. c) Extending X, Y , Z to global left-invariant vector fields, give SU(2) the metric g(X; X) = g(Y; Y ) = g(Z; Z) = 1 and all other inner products zero. Show this is a bi-invariant metric. d) Pick > 0 and set g(X; X) = 2, leaving g(Y; Y ) = g(Z; Z) = 1. Show this is left-invariant but not bi-invariant. p 2) The realification of an n × n complex matrix A + −1B is its assignment it to the 2n × 2n matrix A −B (3) BA Any n × n quaternionic matrix can be written A + Bk where A and B are complex matrices. Its complexification is the 2n × 2n complex matrix A −B (4) B A a) Show that the realification of complex matrices and complexifica- tion of quaternionic matrices are algebra homomorphisms.
    [Show full text]
  • Math 223 Symmetric and Hermitian Matrices. Richard Anstee an N × N Matrix Q Is Orthogonal If QT = Q−1
    Math 223 Symmetric and Hermitian Matrices. Richard Anstee An n × n matrix Q is orthogonal if QT = Q−1. The columns of Q would form an orthonormal basis for Rn. The rows would also form an orthonormal basis for Rn. A matrix A is symmetric if AT = A. Theorem 0.1 Let A be a symmetric n × n matrix of real entries. Then there is an orthogonal matrix Q and a diagonal matrix D so that AQ = QD; i.e. QT AQ = D: Note that the entries of M and D are real. There are various consequences to this result: A symmetric matrix A is diagonalizable A symmetric matrix A has an othonormal basis of eigenvectors. A symmetric matrix A has real eigenvalues. Proof: The proof begins with an appeal to the fundamental theorem of algebra applied to det(A − λI) which asserts that the polynomial factors into linear factors and one of which yields an eigenvalue λ which may not be real. Our second step it to show λ is real. Let x be an eigenvector for λ so that Ax = λx. Again, if λ is not real we must allow for the possibility that x is not a real vector. Let xH = xT denote the conjugate transpose. It applies to matrices as AH = AT . Now xH x ≥ 0 with xH x = 0 if and only if x = 0. We compute xH Ax = xH (λx) = λxH x. Now taking complex conjugates and transpose (xH Ax)H = xH AH x using that (xH )H = x. Then (xH Ax)H = xH Ax = λxH x using AH = A.
    [Show full text]
  • Inner Product Spaces
    CHAPTER 6 Woman teaching geometry, from a fourteenth-century edition of Euclid’s geometry book. Inner Product Spaces In making the definition of a vector space, we generalized the linear structure (addition and scalar multiplication) of R2 and R3. We ignored other important features, such as the notions of length and angle. These ideas are embedded in the concept we now investigate, inner products. Our standing assumptions are as follows: 6.1 Notation F, V F denotes R or C. V denotes a vector space over F. LEARNING OBJECTIVES FOR THIS CHAPTER Cauchy–Schwarz Inequality Gram–Schmidt Procedure linear functionals on inner product spaces calculating minimum distance to a subspace Linear Algebra Done Right, third edition, by Sheldon Axler 164 CHAPTER 6 Inner Product Spaces 6.A Inner Products and Norms Inner Products To motivate the concept of inner prod- 2 3 x1 , x 2 uct, think of vectors in R and R as x arrows with initial point at the origin. x R2 R3 H L The length of a vector in or is called the norm of x, denoted x . 2 k k Thus for x .x1; x2/ R , we have The length of this vector x is p D2 2 2 x x1 x2 . p 2 2 x1 x2 . k k D C 3 C Similarly, if x .x1; x2; x3/ R , p 2D 2 2 2 then x x1 x2 x3 . k k D C C Even though we cannot draw pictures in higher dimensions, the gener- n n alization to R is obvious: we define the norm of x .x1; : : : ; xn/ R D 2 by p 2 2 x x1 xn : k k D C C The norm is not linear on Rn.
    [Show full text]
  • Chapter 1: Complex Numbers Lecture Notes Math Section
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Almae Matris Studiorum Campus Chapter 1: Complex Numbers Lecture notes Math Section 1.1: Definition of Complex Numbers Definition of a complex number A complex number is a number that can be expressed in the form z = a + bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation i2 = −1. In this expression, a is the real part Re(z) and b is the imaginary part Im(z) of the complex number. The complex number a + bi can be identified with the point (a; b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. Ex.1 Understanding complex numbers Write the real part and the imaginary part of the following complex numbers and plot each number in the complex plane. (1) i (2) 4 + 2i (3) 1 − 3i (4) −2 Section 1.2: Operations with Complex Numbers Addition and subtraction of two complex numbers To add/subtract two complex numbers we add/subtract each part separately: (a + bi) + (c + di) = (a + c) + (b + d)i and (a + bi) − (c + di) = (a − c) + (b − d)i Ex.1 Addition and subtraction of complex numbers (1) (9 + i) + (2 − 3i) (2) (−2 + 4i) − (6 + 3i) (3) (i) − (−11 + 2i) (4) (1 + i) + (4 + 9i) Multiplication of two complex numbers To multiply two complex numbers we proceed as follows: (a + bi)(c + di) = ac + adi + bci + bdi2 = ac + adi + bci − bd = (ac − bd) + (ad + bc)i Ex.2 Multiplication of complex numbers (1) (3 + 2i)(1 + 7i) (2) (i + 1)2 (3) (−4 + 3i)(2 − 5i) 1 Chapter 1: Complex Numbers Lecture notes Math Conjugate of a complex number The complex conjugate of the complex number z = a + bi is defined to be z¯ = a − bi.
    [Show full text]
  • COMPLEX MATRICES and THEIR PROPERTIES Mrs
    ISSN: 2277-9655 [Devi * et al., 6(7): July, 2017] Impact Factor: 4.116 IC™ Value: 3.00 CODEN: IJESS7 IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY COMPLEX MATRICES AND THEIR PROPERTIES Mrs. Manju Devi* *Assistant Professor In Mathematics, S.D. (P.G.) College, Panipat DOI: 10.5281/zenodo.828441 ABSTRACT By this paper, our aim is to introduce the Complex Matrices that why we require the complex matrices and we have discussed about the different types of complex matrices and their properties. I. INTRODUCTION It is no longer possible to work only with real matrices and real vectors. When the basic problem was Ax = b the solution was real when A and b are real. Complex number could have been permitted, but would have contributed nothing new. Now we cannot avoid them. A real matrix has real coefficients in det ( A - λI), but the eigen values may complex. We now introduce the space Cn of vectors with n complex components. The old way, the vector in C2 with components ( l, i ) would have zero length: 12 + i2 = 0 not good. The correct length squared is 12 + 1i12 = 2 2 2 2 This change to 11x11 = 1x11 + …….. │xn│ forces a whole series of other changes. The inner product, the transpose, the definitions of symmetric and orthogonal matrices all need to be modified for complex numbers. II. DEFINITION A matrix whose elements may contain complex numbers called complex matrix. The matrix product of two complex matrices is given by where III. LENGTHS AND TRANSPOSES IN THE COMPLEX CASE The complex vector space Cn contains all vectors x with n complex components.
    [Show full text]
  • MATH 304 Linear Algebra Lecture 25: Complex Eigenvalues and Eigenvectors
    MATH 304 Linear Algebra Lecture 25: Complex eigenvalues and eigenvectors. Orthogonal matrices. Rotations in space. Complex numbers C: complex numbers. Complex number: z = x + iy, where x, y R and i 2 = 1. ∈ − i = √ 1: imaginary unit − Alternative notation: z = x + yi. x = real part of z, iy = imaginary part of z y = 0 = z = x (real number) ⇒ x = 0 = z = iy (purely imaginary number) ⇒ We add, subtract, and multiply complex numbers as polynomials in i (but keep in mind that i 2 = 1). − If z1 = x1 + iy1 and z2 = x2 + iy2, then z1 + z2 = (x1 + x2) + i(y1 + y2), z z = (x x ) + i(y y ), 1 − 2 1 − 2 1 − 2 z z = (x x y y ) + i(x y + x y ). 1 2 1 2 − 1 2 1 2 2 1 Given z = x + iy, the complex conjugate of z is z¯ = x iy. The modulus of z is z = x 2 + y 2. − | | zz¯ = (x + iy)(x iy) = x 2 (iy)2 = x 2 +py 2 = z 2. − − | | 1 z¯ 1 x iy z− = ,(x + iy) = − . z 2 − x 2 + y 2 | | Geometric representation Any complex number z = x + iy is represented by the vector/point (x, y) R2. ∈ y r φ 0 x 0 x = r cos φ, y = r sin φ = z = r(cos φ + i sin φ) = reiφ ⇒ iφ1 iφ2 If z1 = r1e and z2 = r2e , then i(φ1+φ2) i(φ1 φ2) z1z2 = r1r2e , z1/z2 = (r1/r2)e − . Fundamental Theorem of Algebra Any polynomial of degree n 1, with complex ≥ coefficients, has exactly n roots (counting with multiplicities).
    [Show full text]
  • The Integration Problem for Complex Lie Algebroids
    The integration problem for complex Lie algebroids Alan Weinstein∗ Department of Mathematics University of California Berkeley CA, 94720-3840 USA [email protected] July 17, 2018 Abstract A complex Lie algebroid is a complex vector bundle over a smooth (real) manifold M with a bracket on sections and an anchor to the complexified tangent bundle of M which satisfy the usual Lie algebroid axioms. A proposal is made here to integrate analytic complex Lie al- gebroids by using analytic continuation to a complexification of M and integration to a holomorphic groupoid. A collection of diverse exam- ples reveal that the holomorphic stacks presented by these groupoids tend to coincide with known objects associated to structures in com- plex geometry. This suggests that the object integrating a complex Lie algebroid should be a holomorphic stack. 1 Introduction It is a pleasure to dedicate this paper to Professor Hideki Omori. His work arXiv:math/0601752v1 [math.DG] 31 Jan 2006 over many years, introducing ILH manifolds [30], Weyl manifolds [32], and blurred Lie groups [31] has broadened the notion of what constitutes a “space.” The problem of “integrating” complex vector fields on real mani- folds seems to lead to yet another kind of space, which is investigated in this paper. ∗research partially supported by NSF grant DMS-0204100 MSC2000 Subject Classification Numbers: Keywords: 1 Recall that a Lie algebroid over a smooth manifold M is a real vector bundle E over M with a Lie algebra structure (over R) on its sections and a bundle map ρ (called the anchor) from E to the tangent bundle T M, satisfying the Leibniz rule [a, fb]= f[a, b] + (ρ(a)f)b for sections a and b and smooth functions f : M → R.
    [Show full text]
  • Complex Numbers and Coordinate Transformations WHOI Math Review
    Complex Numbers and Coordinate transformations WHOI Math Review Isabela Le Bras July 27, 2015 Class outline: 1. Complex number algebra 2. Complex number application 3. Rotation of coordinate systems 4. Polar and spherical coordinates 1 Complex number algebra Complex numbers are ap combination of real and imaginary numbers. Imaginary numbers are based around the definition of i, i = −1. They are useful for solving differential equations; they carry twice as much information as a real number and there exists a useful framework for handling them. To add and subtract complex numbers, group together the real and imaginary parts. For example, (4 + 3i) + (3 + 2i) = 7 + 5i: Try a few examples: • (9 + 3i) - (4 + 7i) = • (6i) + (8 + 2i) = • (7 + 7i) - (9 - 9i) = To multiply, multiply all components by each other, and use the fact that i2 = −1 to simplify. For example, (3 − 2i)(4 + 3i) = 12 + 9i − 8i − 6i2 = 18 + i To divide, multiply the numerator by the complex conjugate of the denominator. The complex conjugate is formed by multiplying the imaginary part of the complex number by -1, and is often denoted by a star, i.e. (6 + 3i)∗ = (6 − 3i). The reason this works is easier to see in the complex plane. Here is an example of division: (4 + 2i)=(3 − i) = (4 + 2i)(3 + i) = 12 + 4i + 6i + 2i2 = 10 + 10i Try a few examples of multiplication and division: • (4 + 7i)(2 + i) = 1 • (5 + 3i)(5 - 2i) = • (6 -2i)/(4 + 3i) = • (3 + 2i)/(6i) = We often think of complex numbers as living on the plane of real and imaginary numbers, and can write them as reiq = r(cos(q) + isin(q)).
    [Show full text]
  • C*-Algebras, Classification, and Regularity
    C*-algebras, classification, and regularity Aaron Tikuisis [email protected] University of Aberdeen Aaron Tikuisis C*-algebras, classification, and regularity C*-algebras The canonical C*-algebra is B(H) := fcontinuous linear operators H ! Hg; where H is a (complex) Hilbert space. B(H) is a (complex) algebra: multiplication = composition of operators. Operators are continuous if and only if they are bounded (operator) norm on B(H). ∗ Every operator T has an adjoint T involution on B(H). A C*-algebra is a subalgebra of B(H) which is norm-closed and closed under adjoints. Aaron Tikuisis C*-algebras, classification, and regularity C*-algebras The canonical C*-algebra is B(H) := fcontinuous linear operators H ! Hg; where H is a (complex) Hilbert space. B(H) is a (complex) algebra: multiplication = composition of operators. Operators are continuous if and only if they are bounded (operator) norm on B(H). ∗ Every operator T has an adjoint T involution on B(H). A C*-algebra is a subalgebra of B(H) which is norm-closed and closed under adjoints. Aaron Tikuisis C*-algebras, classification, and regularity C*-algebras The canonical C*-algebra is B(H) := fcontinuous linear operators H ! Hg; where H is a (complex) Hilbert space. B(H) is a (complex) algebra: multiplication = composition of operators. Operators are continuous if and only if they are bounded (operator) norm on B(H). ∗ Every operator T has an adjoint T involution on B(H). A C*-algebra is a subalgebra of B(H) which is norm-closed and closed under adjoints. Aaron Tikuisis C*-algebras, classification, and regularity C*-algebras The canonical C*-algebra is B(H) := fcontinuous linear operators H ! Hg; where H is a (complex) Hilbert space.
    [Show full text]
  • Commutative Quaternion Matrices for Its Applications
    Commutative Quaternion Matrices Hidayet H¨uda KOSAL¨ and Murat TOSUN November 26, 2016 Abstract In this study, we introduce the concept of commutative quaternions and commutative quaternion matrices. Firstly, we give some properties of commutative quaternions and their Hamilton matrices. After that we investigate commutative quaternion matrices using properties of complex matrices. Then we define the complex adjoint matrix of commutative quaternion matrices and give some of their properties. Mathematics Subject Classification (2000): 11R52;15A33 Keywords: Commutative quaternions, Hamilton matrices, commu- tative quaternion matrices. 1 INTRODUCTION Sir W.R. Hamilton introduced the set of quaternions in 1853 [1], which can be represented as Q = {q = t + xi + yj + zk : t,x,y,z ∈ R} . Quaternion bases satisfy the following identities known as the i2 = j2 = k2 = −1, ij = −ji = k, jk = −kj = i, ki = −ik = j. arXiv:1306.6009v1 [math.AG] 25 Jun 2013 From these rules it follows immediately that multiplication of quaternions is not commutative. So, it is not easy to study quaternion algebra problems. Similarly, it is well known that the main obstacle in study of quaternion matrices, dating back to 1936 [2], is the non-commutative multiplication of quaternions. In this sense, there exists two kinds of its eigenvalue, called right eigenvalue and left eigenvalue. The right eigenvalues and left eigenvalues of quaternion matrices must be not the same. The set of right eigenvalues is always nonempty, and right eigenvalues are well studied in literature [3]. On the contrary, left eigenvalues are less known. Zhang [4], commented that the set of left eigenvalues is not easy to handle, and few result have been obtained.
    [Show full text]
  • Unitary-And-Hermitian-Matrices.Pdf
    11-24-2014 Unitary Matrices and Hermitian Matrices Recall that the conjugate of a complex number a + bi is a bi. The conjugate of a + bi is denoted a + bi or (a + bi)∗. − In this section, I’ll use ( ) for complex conjugation of numbers of matrices. I want to use ( )∗ to denote an operation on matrices, the conjugate transpose. Thus, 3 + 4i = 3 4i, 5 6i =5+6i, 7i = 7i, 10 = 10. − − − Complex conjugation satisfies the following properties: (a) If z C, then z = z if and only if z is a real number. ∈ (b) If z1, z2 C, then ∈ z1 + z2 = z1 + z2. (c) If z1, z2 C, then ∈ z1 z2 = z1 z2. · · The proofs are easy; just write out the complex numbers (e.g. z1 = a+bi and z2 = c+di) and compute. The conjugate of a matrix A is the matrix A obtained by conjugating each element: That is, (A)ij = Aij. You can check that if A and B are matrices and k C, then ∈ kA + B = k A + B and AB = A B. · · You can prove these results by looking at individual elements of the matrices and using the properties of conjugation of numbers given above. Definition. If A is a complex matrix, A∗ is the conjugate transpose of A: ∗ A = AT . Note that the conjugation and transposition can be done in either order: That is, AT = (A)T . To see this, consider the (i, j)th element of the matrices: T T T [(A )]ij = (A )ij = Aji =(A)ji = [(A) ]ij. Example. If 1 2i 4 1 + 2i 2 i 3i ∗ − A = − , then A = 2 + i 2 7i .
    [Show full text]
  • Lecture No 9: COMPLEX INNER PRODUCT SPACES (Supplement) 1
    Lecture no 9: COMPLEX INNER PRODUCT SPACES (supplement) 1 1. Some Facts about Complex Numbers In this chapter, complex numbers come more to the forefront of our discussion. Here we give enough background in complex numbers to study vector spaces where the scalars are the complex numbers. Complex numbers, denoted by ℂ, are defined by where 푖 = √−1. To plot complex numbers, one uses a two-dimensional grid Complex numbers are usually represented as z, where z = a + i b, with a, b ℝ. The arithmetic operations are what one would expect. For example, if If a complex number z = a + ib, where a, b R, then its complex conjugate is defined as 푧̅ = 푎 − 푖푏. 1 9 Л 2 Комплексний Евклідов простір. Аксіоми комплексного скалярного добутку. 10 ПР 2 Обчислення комплексного скалярного добутку. The following relationships occur Also, z is real if and only if 푧̅ = 푧. If the vector space is ℂn, the formula for the usual dot product breaks down in that it does not necessarily give the length of a vector. For example, with the definition of the Euclidean dot product, the length of the vector (3i, 2i) would be √−13. In ℂn, the length of the vector (a1, …, an) should be the distance from the origin to the terminal point of the vector, which is 2. Complex Inner Product Spaces This section considers vector spaces over the complex field ℂ. Note: The definition of inner product given in Lecture 6 is not useful for complex vector spaces because no nonzero complex vector space has such an inner product.
    [Show full text]