# Geršgorin and His Circles

Springer Science & Business Media, Feb 15, 2011 - Mathematics - 230 pages
TheGer? sgorin CircleTheorem, averywell-known resultin linear algebra today, stems from the paper of S. Ger? sgorin in 1931 (which is reproduced in AppendixD)where,givenanarbitraryn×ncomplexmatrix,easyarithmetic operationsontheentriesofthematrixproducendisks,inthecomplexplane, whose union contains all eigenvalues of the given matrix. The beauty and simplicity of Ger? sgorin’s Theorem has undoubtedly inspired further research in this area, resulting in hundreds of papers in which the name “Ger? sgorin” appears. The goal of this book is to give a careful and up-to-date treatment of various aspects of this topic. The author ?rst learned of Ger? sgorin’s results from friendly conversations with Olga Taussky-Todd and John Todd, which inspired me to work in this area.Olgawasclearlypassionateaboutlinearalgebraandmatrixtheory,and her path-?nding 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 Ger? sgorin’s Theorem, will be of interest to many. This book is a?ectionately 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 ?rst recurring theme is that a nonsingularity theorem for a mat- ces 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 Ger? sgorin’s paper appeared. That these two items, nonsingularity theorems and eigenvalue inclusion sets, go hand-in-hand, will be often seen in this book.

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### Contents

 1 Basic Theory 1 2 GerˇsgorinType Eigenvalue InclusionTheorems 35 3 More Eigenvalue Inclusion Results 73 4 Minimal Gerˇsgorin Sets and TheirSharpness 97 5 GFunctions 127 6 GerˇsgorinType Theorems for PartitionedMatrices 155 Appendix A Gerˇsgorins Paper from 1931and Comments on His Life and Research 189
 Appendix B Vector Norms and InducedOperator Norms 199 Appendix C The PerronFrobenius Theory ofNonnegative Matrices MMatrices andHMatrices 201 Appendix D Matlab 6 Programs 205 References 217 Index 223 Symbol Index 225 Copyright