Problems and solutions in introductory and advanced matrix calculus

566 Pages·2017·11.24 MB·English

by  Hardy

Willi-Hans Steeb
Yorick Hardy
PROBLEMS
AND
SOLUTIONS
IN
INTRODUCTORY
AND
ADVANCED MATRIX
CALCULUS
Second Edition —
World Scientific
PROBLEMS
AND
SOLUTIONS
IN
INTRODUCTORY
AND
ADVANCED MATRIX
CALCULUS
— Second Edition —
This page intentionally left blank
Willi-Hans Steeb
Yorick Hardy
University of Johannesburgf South Africa &
University of South Africa, South Africa
PROBLEMS
AND
SOLUTIONS
IN
INTRODUCTORY
AND
ADVANCED MATRIX
CALCULUS
— Second Edition —
O World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING - SHANGHAI • HONGKONG • TAIPEI • CHENNAI* TOKYO
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
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, 2016. |
Includes bibliographical references and index.
Identifiers: LCCN 2016028706| ISBN 9789813143784 (hardcover : alk. paper) |
ISBN 9789813143791 (pbk. : alk. paper)
Subjects: LCSH: Matrices--Problems, exercises, etc. | Calculus. | Mathematical physics.
Classification: LCC QA188 .S664 2016 | DDC 512.9/434~dc23
LC record available at https://lccn.loc.gov/2016028706
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,
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the publisher.
Printed in Singapore
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 re­
search 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 im­
ages. 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 sup­
plementary 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
Note to the Readers
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.
e-mail addresses of the first author:
SteebwilliQgmail. com
steeb_wh@yahoo. com
e-mail address of the second author:
yorickhardyQgmail. com
Home page of the first author: http: //issc. uj . ac. za
Contents
Preface v
Notation ix
1 Basic Operations I
2 Linear Equations 47
3 Kronecker Product 71
4 Traces, Determinants and Hyperdeterminants 99
5 Eigenvalues and Eigenvectors 142
6 Spectral Theorem 205
7 Commutators and Anticommutators 217
8 Decomposition of Matrices 241
9 Functions of Matrices 260
10 Cayley-Hamilton Theorem 299
11 Hadamard Product 309
12 Norms and Scalar Products 318
13 vec Operator 340
14 Nonnormal Matrices 355
15 Binary Matrices 365
16 Star Product 371
17 Unitary Matrices 377
18 Groups, Lie Groups and Matrices 398
vii
viii Contents
19 Lie Algebras and Matrices 439
20 Braid Group 466
21 Graphs and Matrices 486
22 Hilbert Spaces and Mutually Unbiased Bases 496
23 Linear Differential Equations 507
24 Differentiation and Matrices 520
25 Integration and Matrices 535
Bibliography 547
Index 551
Notation
is defined as
belongs to (a set)
i does not belong to (a set)
n intersection of sets
u union of sets
0 empty set
TcS subset T of set S
SnT the intersection of the sets S and T
SuT the union of the sets S and T
HS) image of set S under mapping /
f 0 9 composition of two mappings (/ o g)(x) = f(g(x))
set of natural numbers
set of natural numbers including 0
set of integers
set of rational numbers
set of real numbers
K+ set of nonnegative real numbers
C set of complex numbers
Kn n-dimensional Euclidean space
space of column vectors with n real components
Cn n-dimensional complex linear space
space of column vectors with n complex components
H Hilbert space
Sn symmetric group on a set of n symbols
»(*) real part of the complex number z
9(*) imaginary part of the complex number z
\z\ modulus of complex number z
\x + iy\ = (x2 + ?/2)1/2, x,y€R
X column vector in Cn
XT transpose of x (row vector)
0 zero (column) vector
norm
x • y = x*y scalar product (inner product) in Cn
x x y vector product in E3
IX