Matrices

Matrix Calculus
and

Kronecker Product
with

Applications
and

C++ Programs

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Matrix Calculus
and

Kronecker Product
with

Applications
and

C++ Programs
Willi-Hans Steeb
International School of Scientific Computing, Rand Afrikaans University in collaboration with

Tan Kiat Shi
National University of Singapore

Singapore • NewJersey 'London 'Hong Kong

WorM Scientific

Published by World Scientific Publishing Co. Pte. Ltd. PO Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Steeb, W.-H. Matrix calculus and Kronecker product with applications and C++programs / Willi-Hans Steeb, in collaboration with Tan Kiat Shi. p. cm. Includes bibliographical references and indexes. ISBN 9810232411 1. Matrices. 2. Kronecker products. 3. Matrices--Data processing. 4. Kronecker products ■- Data processing. 5. C++ (Computer program language) I Shi, Tart Kiat. II Title. QA188.S663 1997 530.15?9434-dc21 97-26420 CIP

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

Copyright < 1997 by World Scientific Publishing Co. Pte. Ltd. D All rights reserved. This book, ,r pans thereof, may not be beproduced in iny form or by ana means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without wriitenpermission from the Publisherr

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This book is printed on acid-free paper

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Preface
The Kronecker product of matrices plays animportant role in mathematics and in ap­ plications found in theoretical physics. Such applications are signal processing where the Fourier and Hadamard matrices play the central role. In group theory and ma­ trix representation theory the Kronecker product also comes into play. In statistical mechanics we apply the Kronecker product in the calculation of the partition function and free energy ofspin and Fermi systems. Furthermore the spectral theorem for finite dimensional hermitian matrices can be formulated using the Kronecker product. The so-called quantum groups rely heavily on the Kronecker product. Most books on linear algebra "and matrix theory investigate the Kronecker product only superficially. This book gives a comprehensive introduction to the Kronecker product of matricestogether with its software implementation in C++ using an object-oriented design. In chapter 1 we give a comprehensive introduction into matrix algebra. The basic def­ initions and notations are given in section 1.1. The trace and determinant of square matrices are introduced and their properties are discussed in section 1.2. The eigen­ value problem plays a central role in physics. Section 1.3 isdevoted to this problem. Projection matrices and projection operators are important in Hilbert space theory and quantum mechanics. They are also used in group theoretical reduction in finite group theory. Section 1.4 discusses these matrices. In signal processing Fourier and Hadamard matrices play a central role for the fast Fourier transform and fast Hadamard transform, respectively. Section 1.5 isdevoted to these matrices. Transformations of matrices are described in section 1.6. The invariance of the trance and determinant are also dis­ cussed. Finite groups can be represented as permutation matrices. These matrices are investigated in section 1.7. The vec operator describes an important connection between matrices and vectors. This operator is also important in connection with the...