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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Denis Serre
Matrices Theory and Applications Second Edition
1C
Denis Serre Unité de Mathématiques Pures et Appliquées École Normale Supérieure de Lyon 69364 Lyon Cedex 07 France [email protected] Editorial Board: S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA [email protected]
K. A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720 USA [email protected]
ISSN 0072-5285 ISBN 978-1-4419-7682-6 e-ISBN 978-1-4419-7683-3 DOI 10.1007/978-1-4419-7683-3 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010938321 Mathematics Subject Classification (2010): 15-XX, 22-XX, 47-XX, 65-XX © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, 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 on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To Pascale, Fanny, Paul, and Joachim
Contents
Preface for the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Preface for the First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1
Elementary Linear and Multilinear Algebra . . . . . . . . . . . . . . . . . . . . . . 1.1 Vectors and Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Bilinear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 5 9
2
What Are Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Matrices as Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Matrices and Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15 15 19 28
3
Square Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Minors . . . . . . . . . . . . . . . . . . . .
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