Springer Series in 36 Computational Mathematics

Transcription

1 Springer Series in 36 Computational Mathematics Editorial Board R. Bank, La Jolla, CA R.L. Graham, La Jolla, CA J. Stoer, Würzburg R. Varga, Kent, Ohio H. Yserentant, Berlin

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3 Richard S. Varga Geršgorin and His Circles 123

4 Richard S. Varga Kent State University Institute for Computational Mathematics Kent,OH44242,USA Cover jacket figure: Wassily Kandinsky, Heavy Circles, Oil on canvas. Norton Simon Museum, The Blue Four Galka Scheyer Collection VG Bild-Kunst, Bonn 2004 Library of Congress Control Number: Mathematics Subject Classification (2000): 15A18, 15A42, 15A48, 15A60, 65F15, 65F35 ISSN ISBN SpringerBerlinHeidelbergNewYork This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks.duplicationofthispublicationorpartsthereofispermittedonlyundertheprovisions of the German Copyright Law of September 9, 1965, initscurrentversion, andpermissionfor use must always be obtained from Springer-Verlag. Violations are liable to prosecution under thegermancopyrightlaw. Springer is a part of Springer Science+Business Media springeronline.com Springer-Verlag Berlin Heidelberg 2004 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: design&production,heidelberg Typeset by the author using a Springer L A TEX macro package Printed on acid-free paper 46/3142LK

5 Dedicated to: Olga Taussky-Todd and John Todd Department of Mathematics California Institute of Technology Pasadena, California Olga died on October 7, 1995, in Pasadena, California.

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7 I. Preface The Geršgorin Circle Theorem, a very well-known result in linear algebra today, stems from the paper of S. Geršgorin in 1931 (which is reproduced in Appendix D) where, given an arbitrary n n complex matrix, easy arithmetic operations on the entries of the matrix produce n disks, in the complex plane, whose union contains all eigenvalues of the given matrix. The beauty and simplicity of Geršgorin s Theorem has undoubtedly inspired further research in this area, resulting in hundreds of papers in which the name Geršgorin appears. The goal of this book is to give a careful and up-to-date treatment of various aspects of this topic. The author first learned of Geršgorin s results from friendly conversations with Olga Taussky-Todd and John Todd, which inspired me to work in this area. Olga was clearly passionate about linear algebra and matrix theory, and her path-finding results in these areas were like a magnet to many, including this author! It is the author s hope that the results, presented here on topics related to Geršgorin s Theorem, will be of interest to many. This book is affectionately dedicated to my mentors, Olga Taussky-Todd and John Todd. There are two main recurring themes which the reader will see in this book. The first recurring theme is that a nonsingularity theorem for a matrices gives rise to an equivalent eigenvalue inclusion set in the complex plane for matrices, and conversely. Though common knowledge today, this was not widely recognized until many years after Geršgorin s paper appeared. That these two items, nonsingularity theorems and eigenvalue inclusion sets, go hand-in-hand, will be often seen in this book. The second recurring theme in this book is the decisive role that M- matrices and H-matrices, and the related Perron-Frobenius theory of nonnegative matrices, play throughout this book. A much lesser recurring theme in this book is the observation that there have been surprisingly many published results in this area which contain major errors. Most of these errors have fortunately been corrected in subsequent papers. As the reader will see, there are many new and unpublished items in the six chapters of this book. This is the result of trying to collect and unify many Geršgorin-type results in the literature, and it has been truly exciting to work in this area. The material in this book is almost entirely self-contained, with exercises for those who wish to test their skills. The material in this

8 VIII I. Preface book should be within the grasp of upper-class undergraduates and graduate students in mathematics, as well as physicists, engineers, and computer scientists. But there are a number of related topics which interested readers may find missing in these chapters. These topics include infinite dimensional extensions of Geršgorin s theory, connections with pseudo-spectra and singular values of matrices, real eigenvalues of real matrices, and generalized matrix eigenvalue problems. The results in these areas are currently, in the author s opinion, more fragmented and less unified, so that their complete coverage is intended for a future second volume of this book, with the same title. To this end, the author warmly solicits suggestions and references for this second volume, as well as comments, and corrections on this present volume. (Please send to varga@math.kent.edu.) A large number of people have read parts of this manuscript and have offered many worthwhile suggestions to the author; I owe them my heartfelt thanks. But, there are a few whose contributions here must be fully recognized. My sincere thanks are due to my colleagues, Arden Ruttan and Laura Smithies. Arden assembled the Matlab programs in Appendix D for many of the pictures in this book, and his careful work identified some missing eigenvalues in Fig. 6.2, 6.3, 6.5, 6.6. Laura read the entire manuscript and set for herself the daunting task of working through all the exercises given in this book! My greatest thanks go to experts in this area, Richard Brualdi (University of Wisconsin), Ljiljana Cvetković (University of Novi-Sad), Volker Mehrmann (Technische Universität-Berlin), François Robert (University of Grenoble), and anonymous referees, who have made substantial comments and suggestions on the entire manuscript. It is a pleasure for me to recognize their deep expertise in this area and their unselfish efforts to help the author with their comments. Our warmest and sincere thanks are due to our secretaries (in the Institute for Computational Mathematics at Kent State University) Joyce Fuell, who began the typing of this manuscript, and to Misty Tackett, who completed, with great care, the typing of this manuscript, and to my wife, Esther, for her love, understanding and great forbearance in the writing of the book. Finally, we wish to thank Springer-Verlag for their strong and unfailing support over the years, of the author s efforts. Richard S. Varga March 3, 2004 Kent, Ohio

9 Contents I. Preface... VII 1. Basic Theory Geršgorin stheorem Extensions of Geršgorin s Theorem via Graph Theory Analysis Extensions of Geršgorin s Theorem and Fan s Theorem A Norm Derivation of Geršgorin s Theorem Geršgorin-Type Eigenvalue Inclusion Theorems Brauer sovalsofcassini Higher-OrderLemniscates Comparison of the Brauer Sets and the Brualdi Sets The Sharpness of Brualdi Lemniscate Sets AnExample More Eigenvalue Inclusion Results The Parodi-Schneider Eigenvalue Inclusion Sets TheFieldofValuesofaMatrix Newer Eigenvalue Inclusion Sets The Pupkov-Solov ev Eigenvalue Inclusions Set Minimal Geršgorin Sets and Their Sharpness Minimal Geršgorin Sets Minimal Geršgorin SetsviaPermutations A Comparison of Minimal Geršgorin Sets and Brualdi Sets G-Functions The Sets F n and G n Structural Properties of G n and Gn c Minimal G-Functions Minimal G-Functions with Small Domains of Dependence Connections with Brauer Sets and Generalized Brualdi Sets.. 149

10 X Contents 6. Geršgorin-Type Theorems for Partitioned Matrices Partitioned Matrices and Block Diagonal Dominance A Different Norm Approach A Variation on a Theme by Brualdi G-FunctionsinthePartitionedCase Appendix A. Geršgorin s Paper from 1931, and Comments Appendix B. Vector Norms and Induced Operator Norms Appendix C. The Perron-Frobenius Theory of Nonnegative Matrices Appendix D. Matlab 6 Programs References