Toeplitz matrices, asymptotic linear algebra, and functional analysisToeplitz矩阵、渐进线性代数,和功能分析

作者： Albrecht Böttcher，Sergei M. Grudsky著

出 版 社：

出版时间： 2000-7-1字数：版次：页数： 116印刷时间： 2000/07/01开本： 16开印次：纸张： 胶版纸I S B N ： 9783764362904包装： 平装内容简介

This text is a self-contained introduction to some problems for Toeplitz matrices that are placed in the borderland between linear algebra and functional analysis. The text looks at Toeplitz matrices with rational symbols, and focuses attention on the asymptotic behavior of the singular values, which includes the behavior of the norms, the norms of the inverses, and the condition numbers as special cases. The text illustrates that the asymptotics of several linear algebra characteristics depend in a fascinating way on functional analytic properties of infinite matrices. Many convergence results can very comfortably be obtained by working with appropriate C*-algebras, while refinements of these results, for example, estimates of the convergence speed, nevertheless require hard analysis.

目录

Preface

1 Infinite Toeplitz Matrices

1.1 Boundedness

1.2 Compactness and Selfadjointness

1.3 C*-Algebras

1.4 Fredhohn Operators

1.5 Continuous Symbols

1.6 Notes

2 C*-Algebras in Action

2.1 Finite Section Method

2.2 Perturbed Toeplitz Matrices

2.3 Algebraization of Stability

2.4 Stability Criteria

2.5 Asymptotic Inverses

2.6 Norms

2.7 Norms of Inverses

2.8 Condition Numbers

2.9 Eigenvalues of Hermitian Matrices

2.10 Singular Values

2.11 Pseudospectra

2.12 Notes

3 Instability

3.1 Some Standard Tricks

3.2 At Least Polynomial Growth

3.3 At Most Polynomial Growth

3.4 Semi-Definite Hermitian Matrices

3.5 Exponential Growth

3.6 Notes

4 Condition Numbers

4.1 Norms of Pure Toeplitz Matrices

4.2 Structure of the Inverse

4.3 Perturbed Toeplitz Band Matrices

4.4 Norms of the Inverses of Toeplitz Matrices

4.5 Condition Numbers of Toeplitz Matrices

4.6 Condition Numbers of Toeplitz-Like Matrices

4.7 Notes

5 Singular Values

5.1 Basic Facts

5.2 The Splitting Phenomenon

5.3 Extreme Singular Values

5.4 The First Szeg5 Limit Theorem

5.5 The Avram-Parter Theorem

5.6 Circulants

5.7 Toeplitz Band Matrices

5.8 Zizler, Zuidwijk, Taylor, Arimoto

5.9 Notes

Bibliography

Notation Index

Subject Index