Gerschgorin and his circlesGerschgorin和他的圈子
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作者： Richard S. Varga，Richard Varga著
出 版 社： 北京燕山出版社
出版时间： 2004-10-1字数：版次： 1页数： 226印刷时间： 2004/10/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9783540211006包装： 精装内容简介
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. Since the publication of this paper, there has been many newer results spawned by his paper, and this book will be the first which is devoted solely to this resulting area. As such, it will include the latest research results, such as Brauer ovals of Cassini and Brualdi lemniscates, and their comparisons. This book is dedicated to the late Olga Taussky-Todd and her husband, John Todd. It was Olga who brought to light Geršgorin's paper and its significance to the mathematical world. The level of this book requires only a modest background in linear algebra and analysis, and is therefore comprehensible to upper-level and graduate level students in mathematics.
目录
I. Preface
2Basic Theory
1.1 Gersgorin's Theorem
1.2 Extensions of Gersgorin's Theorem via Graph Theory
1.3 Analysis Extensions of Gersgorin's Theorem and Fan's Theorem
1.4 A Norm Derivation of Gersgorin's Theorem 1.1
1Gersgorin-Type Eigenvalue Inclusion Theorems
2.1 Brauer's Ovals of Cassini
2.2 Higher-Order Lemniscates
2.3 Comparison of the Brauer Sets and the Brualdi Sets
2.4 The Sharpness of Brualdi Lemniscate Sets
2.5 An Example
3More Eigenvalue Inclusion Results
3.1 The Parodi-Schneider Eigenvalue Inclusion Sets
3.2 The Field of Values of a Matrix
3.3 Newer Eigenvalue Inclusion Sets
3.4 The Pupkov-Solov'ev Eigenvalue Inclusions Set
4Minimal Gersgorin Sets and Their Sharpness
4.1 Minimal Gersgorin Sets
4.2 Minimal Gersgorin Sets via Permutations
4.3 A Comparison of Minimal Gersgorin Sets and Brualdi Sets
5G-Functions
5.1 The Sets Fn and Gn
5.2 Structural Properties of Gn and Gc
5.3 Minimal G-Functions
5.4 Minimal G-Functions with Small Domains of Dependence
5.5 Connections with Brauer Sets and Generalized Brualdi Sets
6Gersgorin-Type lneorems for Partitioned Matrices
6.1 Partitioned Matrices and Block Diagonal Dominance
6.2 A Different Norm Approach
6.3 A Variation on a Theme by Brualdi
6.4 G-Functions in the Partitioned Case
Appendix A. Gersgorin's Paper from 1931, and Comments
Appendix B. Vector Norms and Induced Operator Norms
Appendix C. The Perron-robemus Theory of Nonnegative Matrices
Appendix D. Matlab 6 Programs
References