Matrices Theory and Applications
In this book, Denis Serre begins by providing a clean and concise introduction to the basic theory of matrices. He then goes on to give many interesting applications of matrices to different aspects of mathematics and also other areas of science and engin
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Editorial Board S. Axler F.W. Gehring K.A. Ribet
Denis Serre
Matrices Theory and Applications
Denis Serre Ecole Normale Supe´rieure de Lyon UMPA Lyon Cedex 07, F-69364 France [email protected]
Editorial Board: S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA [email protected]
F.W. Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA [email protected]
K.A. Ribet Mathematics Department University of California, Berkeley Berkeley, CA 94720-3840 USA [email protected]
Mathematics Subject Classification (2000): 15-01 Library of Congress Cataloging-in-Publication Data Serre, D. (Denis) [Matrices. English.] Matrices : theory and applications / Denis Serre. p. cm.—(Graduate texts in mathematics ; 216) Includes bibliographical references and index. ISBN 0-387-95460-0 (alk. paper) 1. Matrices I. Title. II. Series. QA188 .S4713 2002 512.9′434—dc21 2002022926 ISBN 0-387-95460-0
Printed on acid-free paper.
Translated from Les Matrices: The´orie et pratique, published by Dunod (Paris), 2001. © 2002 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1
SPIN 10869456
Typesetting: Pages created by the author in LaTeX2e. www.springer-ny.com Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH
To Pascale and Joachim
Preface
The study of matrices occupies a singular place within mathematics. It is still an area of active research, and it is used by every mathematician and by many scientists working in various specialities. Several examples illustrate its versatility: • Scientific computing libraries began growing around matrix calculus. As a matter of fact, the discretization of partial differential operators is an endless source of linear finite-dimensional problems. • At a discrete level, the maximum principle is related to nonnegative matrices. • Control theory and stabilization of systems with finitely many degrees of freedom involve spectral analysis of matrices. • The discrete Fourier transform, including the fast Fourier transform, makes use of Toeplitz matrices. • Statistics is widely based on correlation matrices. • The generalized inverse is involved in least-squares approximation. • Symmetric matrices are inertia,
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