Problems and Solutions in Introductory and Advanced Matrix Calculus Downloaded from

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4 Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore USA office: 27 Warren Street, Suite , Hackensack, NJ UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Problems and Solutions in Introductory and Advanced Matrix Calculus Downloaded from Library of Congress Cataloging-in-Publication Data Names: Steeb, W.-H. Hardy, Yorick, 1976 Title: Problems and solutions in introductory and advanced matrix calculus. Description: Second edition / by Willi-Hans Steeb (University of Johannesburg, South Africa & University of South Africa, South Africa), Yorick Hardy (University of Johannesburg, South Africa & University of South Africa, South Africa). New Jersey : World Scientific, Includes bibliographical references and index. Identifiers: LCCN ISBN (hardcover : alk. paper) ISBN (pbk. : alk. paper) Subjects: LCSH: Matrices--Problems, exercises, etc. Calculus. Mathematical physics. Classification: LCC QA188.S DDC 512.9/434--dc23 LC record available at British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright 2017 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore

5 Preface The purpose of this book is to supply a collection of problems in introductory and advanced matrix problems together with their detailed solutions which will prove to be valuable to undergraduate and graduate students as well as to research workers in these fields. Each chapter contains an introduction with the essential definitions and explanations to tackle the problems in the chapter. If necessary, other concepts are explained directly with the present problems. Thus the material in the book is self-contained. The topics range in difficulty from elementary to advanced. Students can learn important principles and strategies required for problem solving. Lecturers will also find this text useful either as a supplement or text, since important concepts and techniques are developed in the problems. A large number of problems are related to applications. Applications include wavelets, linear integral equations, Kirchhoff s laws, global positioning systems, Floquet theory, octonians, random walks, entanglement, tensor decomposition, hyperdeterminant, matrix-valued differential forms, Kronecker product and images. A number of problems useful in quantum physics and graph theory are also provided. Advanced topics include groups and matrices, Lie groups and matrices and Lie algebras and matrices. Exercises for matrix-valued differential forms are also included. In this second edition new problems for braid groups, mutually unbiased bases, vec operator, spectral theorem, binary matrices, nonnormal matrices, wavelets, fractals, matrices and integration are added. Each chapter also contains supplementary problems. Furthermore a number of Maxima and SymbolicC++ programs are added for solving problems. Applications in mathematical and theoretical physics are emphasized. The book can also be used as a text for linear and multilinear algebra or matrix theory. The material was tested in the first author s lectures given around the world. v

6 Note to the Readers Problems and Solutions in Introductory and Advanced Matrix Calculus Downloaded from The International School for Scientific Computing (ISSC) provides certificate courses for this subject. Please contact the authors if you want to do this course or other courses of the ISSC. addresses of the first author: address of the second author: Home page of the first author: vi

7 Contents Preface v Notation ix 1 Basic Operations 1 2 Linear Equations 47 3 Kronecker Product 71 4 Traces, Determinants and Hyperdeterminants 99 5 Eigenvalues and Eigenvectors Spectral Theorem Commutators and Anticommutators Decomposition of Matrices Functions of Matrices Cayley-Hamilton Theorem Hadamard Product Norms and Scalar Products vec Operator Nonnormal Matrices Binary Matrices Star Product Unitary Matrices Groups, Lie Groups and Matrices 398 vii

8 viii Contents 19 Lie Algebras and Matrices Braid Group Graphs and Matrices Hilbert Spaces and Mutually Unbiased Bases 496 Problems and Solutions in Introductory and Advanced Matrix Calculus Downloaded from 23 Linear Differential Equations Differentiation and Matrices Integration and Matrices 535 Bibliography 547 Index 551

9 Notation := is defined as belongs to (a set) / does not belong to (a set) intersection of sets union of sets empty set T S subset T of set S S T the intersection of the sets S and T S T the union of the sets S and T f(s) image of set S under mapping f f g composition of two mappings (f g)(x) = f(g(x)) N set of natural numbers N 0 set of natural numbers including 0 Z set of integers Q set of rational numbers R set of real numbers R + set of nonnegative real numbers C set of complex numbers R n n-dimensional Euclidean space space of column vectors with n real components C n n-dimensional complex linear space space of column vectors with n complex components H Hilbert space S n symmetric group on a set of n symbols i 1 R(z) real part of the complex number z I(z) imaginary part of the complex number z z modulus of complex number z x + iy = (x 2 + y 2 ) 1/2, x, y R x column vector in C n x T transpose of x (row vector) 0 zero (column) vector. norm x y x y scalar product (inner product) in C n x y vector product in R 3 ix

10 x Notation Problems and Solutions in Introductory and Advanced Matrix Calculus Downloaded from A, B, C m n matrices P n n permutation matrix Π n n projection matrix U n n unitary matrix vec(a) vectorization of matrix A det(a) determinant of a square matrix A tr(a) trace of a square matrix A Pf(A) Pfaffian of square matrix A rank(a) rank of matrix A A T transpose of matrix A A conjugate of matrix A A conjugate transpose of matrix A A 1 inverse of square matrix A (if it exists) I n n n unit matrix I unit operator 0 n n n zero matrix AB matrix product of m n matrix A and n p matrix B A B Hadamard product (entry-wise product) of m n matrices A and B [A, B] := AB BA commutator for square matrices A and B [A, B] + := AB + BA anticommutator for square matrices A and B A B Kronecker product of matrices A and B A B Direct sum of matrices A and B δ jk Kronecker delta with δ jk = 1 for j = k and δ jk = 0 for j k λ eigenvalue ɛ real parameter t time variable Ĥ Hamilton operator The elementary matrices E jk with j = 1,..., m and k = 1,..., n are defined as 1 at entry (j, k) and 0 otherwise. The Pauli spin matrices are defined as ( ) 0 1 σ 1 :=, σ := ( 0 i i 0 ) ( ) 1 0, σ 3 :=. 0 1