Gersgorin and His Circles

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.

List of contents

Basic Theory.- Gersgorin-Type Eigenvalue Inclusion Theorems.- More Eigenvalue Inclusion Results.- Minimal Gersgorin Sets and Their Sharpness.- G-Functions.- Gersgorin-Type Theorems for Partitioned Matrices.

Summary

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.

Additional text

From the reviews:
"This book is an extremely detailed account of eigenvalue inclusion theorems, starting with the basic Geršgorin disk theorem … . One of the most pleasing features of the book is what Varga calls the first recurring theme: every eigenvalue inclusion theorem has a corresponding nonsingularity theorem. … contains numerous simple examples and illustrative diagrams, and everything is explained in great detail. … For anyone seeking information about eigenvalue inclusion theorems, this book will be a great reference." (David Scott Watkins, Mathematical Reviews, 2005h)
"The monograph presents a survey of results about spectrum localization for matrices, beginning with the first Geršgorin’s eigenvalue inclusion result … . This book contains many new and some unpublished results as a survey of many Geršgorin type results in the literature. The presented material is almost entirely self-contained with exercises which are useful both for graduate students in mathematics and physicists, engineers and computer scientists." (Boris V. Loginov, Zentralblatt MATH, 1057:8, 2005)
"The new book of Varga contains, along with several new results, nearly all that has been discovered about eiegnvalue inclusions since 1931. … One of the merits of this book is the information that it provides about Semën Aronovic Gersgorin … . I like this book and recommend reading it." (L. Elsner, American Mathematical Monthly, Vol. 113, April, 2006)

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From the reviews:
"This book is an extremely detailed account of eigenvalue inclusion theorems, starting with the basic Gersgorin disk theorem ... . One of the most pleasing features of the book is what Varga calls the first recurring theme: every eigenvalue inclusion theorem has a corresponding nonsingularity theorem. ... contains numerous simple examples and illustrative diagrams, and everything is explained in great detail. ... For anyone seeking information about eigenvalue inclusion theorems, this book will be a great reference." (David Scott Watkins, Mathematical Reviews, 2005h)
"The monograph presents a survey of results about spectrum localization for matrices, beginning with the first Gersgorin's eigenvalue inclusion result ... . This book contains many new and some unpublished results as a survey of many Gersgorin type results in the literature. The presented material is almost entirely self-contained with exercises which are useful both for graduate students in mathematics and physicists, engineers and computer scientists." (Boris V. Loginov, Zentralblatt MATH, 1057:8, 2005)
"The new book of Varga contains, along with several new results, nearly all that has been discovered about eiegnvalue inclusions since 1931. ... One of the merits of this book is the information that it provides about Semën Aronovic Gersgorin ... . I like this book and recommend reading it." (L. Elsner, American Mathematical Monthly, Vol. 113, April, 2006)

Product details

• Authors: Richard S Varga, Richard S. Varga
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Released: 31.12.2011
• EAN: 9783540211006
• ISBN: 978-3-540-21100-6
• No. of pages: 230
• Dimensions: 168 mm x 15 mm x 243 mm
• Weight: 496 g
• Illustrations: X, 230 p. 1 illus. in color.
• Series: Springer Series in Computational Mathematics; Springer Series in Computational Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Probability theory, stochastic theory, mathematical statistics; Algebra, B, Mathematics, minimum, Theorem, Mathematics and Statistics, Numerical analysis, Partition, boundary element method, Householder sets, M- and H-matrices, Sharp, minimal Gershgorin sets, Gershgorin disks, G-functions, Robert sets