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Partially Specified Matrices and Operators: Classification, Completion, Applications

  • Book
  • © 1995

Overview

Authors:
  1. Israel Gohberg

    1. School of Mathematical Sciences Raymon and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Israel

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  2. Marinus A. Kaashoek

    1. Faculteit Wiskunde en Informatica, Vrije Universiteit Amsterdam, The Netherlands

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    You can also search for this author in PubMed   Google Scholar

  3. Frederik Schagen

    1. Faculteit Wiskunde en Informatica, Vrije Universiteit Amsterdam, The Netherlands

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Part of the book series: Operator Theory: Advances and Applications (OT, volume 79)

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Table of contents (16 chapters)

  1. Front Matter

    Pages i-viii
    Download chapter PDF

  2. Introduction

    • Israel Gohberg, Marinus A. Kaashoek, Frederik van Schagen
    Pages 1-3

  3. Main Problems and Motivation

    • Israel Gohberg, Marinus A. Kaashoek, Frederik van Schagen
    Pages 5-23

  4. Elementary Operations on Blocks

    • Israel Gohberg, Marinus A. Kaashoek, Frederik van Schagen
    Pages 25-38

  5. Full Length Blocks

    • Israel Gohberg, Marinus A. Kaashoek, Frederik van Schagen
    Pages 39-57

  6. The Eigenvalue Completion Problem for Full Length Blocks

    • Israel Gohberg, Marinus A. Kaashoek, Frederik van Schagen
    Pages 59-82

  7. Full Width Blocks

    • Israel Gohberg, Marinus A. Kaashoek, Frederik van Schagen
    Pages 83-105

  8. Principal Blocks

    • Israel Gohberg, Marinus A. Kaashoek, Frederik van Schagen
    Pages 107-111

  9. General Blocks

    • Israel Gohberg, Marinus A. Kaashoek, Frederik van Schagen
    Pages 113-138

  10. Off-diagonal Blocks

    • Israel Gohberg, Marinus A. Kaashoek, Frederik van Schagen
    Pages 139-142

  11. Connections with Linear Systems

    • Israel Gohberg, Marinus A. Kaashoek, Frederik van Schagen
    Pages 143-176

  12. Applications to Matrix Polynomials

    • Israel Gohberg, Marinus A. Kaashoek, Frederik van Schagen
    Pages 177-188

  13. Applications to Rational Matrix Functions

    • Israel Gohberg, Marinus A. Kaashoek, Frederik van Schagen
    Pages 189-216

  14. Infinite Dimensional Operator Blocks

    • Israel Gohberg, Marinus A. Kaashoek, Frederik van Schagen
    Pages 217-242

  15. Factorization of Operator Polynomials

    • Israel Gohberg, Marinus A. Kaashoek, Frederik van Schagen
    Pages 243-266

  16. Factorization of Analytic Operator Functions

    • Israel Gohberg, Marinus A. Kaashoek, Frederik van Schagen
    Pages 267-276

  17. Eigenvalue Completion Problems for Triangular Matrices

    • Israel Gohberg, Marinus A. Kaashoek, Frederik van Schagen
    Pages 277-295

  18. Back Matter

    Pages 297-336
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Keywords

About this book

This book is devoted to a new direction in linear algebra and operator theory that deals with the invariants of partially specified matrices and operators, and with the spectral analysis of their completions. The theory developed centers around two major problems concerning matrices of which part of the entries are given and the others are unspecified. The first is a classification problem and aims at a simplification of the given part with the help of admissible similarities. The results here may be seen as a far reaching generalization of the Jordan canonical form. The second problem is called the eigenvalue completion problem and asks to describe all possible eigenvalues and their multiplicities of the matrices which one obtains by filling in the unspecified entries. Both problems are also considered in an infinite dimensional operator framework. A large part of the book deals with applications to matrix theory and analysis, namely to stabilization problems in mathematical system theory, to problems of Wiener-Hopf factorization and interpolation for matrix polynomials and rational matrix functions, to the Kronecker structure theory of linear pencils, and to non­ everywhere defined operators. The eigenvalue completion problem has a natural associated inverse, which appears as a restriction problem. The analysis of these two problems is often simpler when a solution of the corresponding classification problem is available.

Authors and Affiliations

  • School of Mathematical Sciences Raymon and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Israel

    Israel Gohberg

  • Faculteit Wiskunde en Informatica, Vrije Universiteit Amsterdam, The Netherlands

    Marinus A. Kaashoek, Frederik Schagen

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