The Schur Complement and Its Applications
The Schur complement plays an important role in matrix analysis, statistics, numerical analysis, and many other areas of mathematics and its applications. This book describes the Schur complement as a rich and basic tool in mathematical research and appli
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Numerical Methods and Algorithms VOLUME 4
Series Editor: Claude Brezinski Universite des Sciences et Technologies de Lille, France
THE SCHUR COMPLEMENT AND ITS APPLICATIONS
Edited by FUZHEN ZHANG Nova Southeastern University, Fort Lauderdale, U.S.A. Shenyang Normal University, Shenyang, China
Spri ringer
Library of Congress Cataloging-in-Publication Data A C.I.P. record for this book is available from the Library of Congress.
ISBN 0-387-24271-6
e-ISBN 0-387-24273-2
Printed on acid-free paper.
© 2005 Springer Science+Business Media, 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 Science+Business Media, Inc., 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 know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the 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 springeronline.com
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To our families, friends, and the matrix community
Issai Schur (1875-1941) This portrait of Issai Schur was apparently made by the "Atelieir Hanni Schwarz, N. W. Dorotheenstrafie 73" in Berlin, c. 1917, and appears in Ausgewdhlte Arbeiten zu den Ursprilngen der Schur-Analysis: Gewidmet dem grofien Mathematiker Issai Schur (1875-1941) edited by Bernd Fritzsche & Bernd Kirstein, pub. B. G. Teubner Verlagsgesellschaft, Stuttgart, 1991.
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/ Emilie Virginia Haynsworth (1916-1985) T h i s p o r t r a i t of E m i h e Virginia H a y n s w o r t h is on t h e A u b u r n University W e b site www.aubum.edu/~fitzpjd/ben/images/Emilie.gif a n d in t h e b o o k The Education of a Mathematician by P h i h p J. Davis, p u b . A K P e t e r s , Natick, Mass., 2000.
Contents
Preface
xv
Chapter 0
Historical Introduction: Issai Schur and the Early Development of the Schur Complement
1
Simo Puntanen, University of Tampere, Tampere, Finland George P. H. Styan, McGill University, Montreal, Canada
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Introduction and mise-en-scene The Schur complement: the name and the notation Some implicit manifestations in the 1800s The lemma and the Schur determinant formula Issai Schur (1875-1941) Schur's contributions in mathematics Publication under J. Schur Boltz 1923, Lohan 1933, Aitken 1937 and the Banchiewicz inversion formula 1937 0.8 Frazer, Duncan & Collar 1938, Aitken 1939, and Duncan 1944 0.9 The Aitken block-diagonalization formula 1939 and the Guttman rank additivity formula 1946 0.10 Emilie Virginia Haynsworth (1916-1985) and the Haynsworth inertia additivity formula
Chapter 1
Basic Properties of the Schur Complement
1 2 3
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