# Geršgorin and His Circles

• Richard S. Varga
Book

Part of the Springer Series in Computational Mathematics book series (SSCM, volume 36)

1. Front Matter
Pages i-x
2. Richard S. Varga
Pages 1-33
3. Richard S. Varga
Pages 35-72
4. Richard S. Varga
Pages 73-96
5. Richard S. Varga
Pages 97-125
6. Richard S. Varga
Pages 127-153
7. Richard S. Varga
Pages 155-187
8. Back Matter
Pages 199-226

### Introduction

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.

### Keywords

Area Brauer ovals of Cassini G-functions Gershgorin disks Householder sets M- and H-matrices Partition Robert sets Sharp algebra boundary element method mathematics minimal Gershgorin sets minimum theorem

#### Authors and affiliations

• Richard S. Varga
• 1
1. 1.Institute for Computational MathematicsKent State UniversityKentUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-17798-9
• Copyright Information Springer-Verlag Berlin Heidelberg 2004
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-540-21100-6
• Online ISBN 978-3-642-17798-9
• Series Print ISSN 0179-3632
• Buy this book on publisher's site
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