Varga R.S. Geršgorin and His Circles

Springer, 2004. — 241 p. — (Springer Series in Computational Mathematics 36). — ISBN: 3540211004.This book studies the original results, and their extensions, of the Russian mathematician S.A. Geršgorin who wrote a seminal paper in 1931 on how to easily obtain estimates of all n eigenvalues (characteristic values) of any given n-by-n complex matrix.
The Gersgorin Circle Theorem, averywell-known resultin linear algebra today, stems from the paper of S. Gersgorin in 1931 (which is reproduced in Appendix D) where, givenanar bitraryn ncomplexmatrix, easyarithmetic operation sontheen triesof them atrix producendisks, inthecomplex plane, whose union contains all eigenvalues of the given matrix. The beauty and simplicity of Gersgorin’s Theorem has undoubtedly inspired further research in this area, resulting in hundreds of papers in which the name “Gersgorin” appears. The goal of this book is to give a careful and up-to-date treatment of various aspects of this topic. The authorst learned of Gersgorin’s results from friendly conversations with Olga Taussky-Todd and John Todd, which inspired me to work in this area.Olgawasclearlypassionateaboutlinearalgebraandmatrixtheory, and her pathinding 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 Gersgorin’s Theorem, will be of interest to many. This book is aectionately 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 Gersgorin’s paper appeared. That these two items, nonsingularity theorems and eigenvalue nclusion sets, go hand-in-hand, will be often seen in this book.Basic Theory
Geršgorin-Type Eigenvalue Inclusion Theorems
More Eigenvalue Inclusion Results
Minimal Geršgorin Sets and Their Sharpness
G-Functions
Geršgorin-Type Theorems for Partitioned Matrices