Ebook: Partially Specified Matrices and Operators: Classification, Completion, Applications

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.

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This book explores a new direction in linear algebra and operator theory dealing 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 problem of classification of partially specified matrices, and the results here may be seen as a far reaching generalization of the Jordan canonical form. The second problem is the eigenvalue completion problem, which asks for a description of 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 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 book will appeal to a wide group of mathematicians and engineers. Much of the material can be used in advanced courses in matrix and operator theory.

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This book explores a new direction in linear algebra and operator theory dealing 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 problem of classification of partially specified matrices, and the results here may be seen as a far reaching generalization of the Jordan canonical form. The second problem is the eigenvalue completion problem, which asks for a description of 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 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 book will appeal to a wide group of mathematicians and engineers. Much of the material can be used in advanced courses in matrix and operator theory.
Content:
Front Matter....Pages i-viii
Introduction....Pages 1-3
Main Problems and Motivation....Pages 5-23
Elementary Operations on Blocks....Pages 25-38
Full Length Blocks....Pages 39-57
The Eigenvalue Completion Problem for Full Length Blocks....Pages 59-82
Full Width Blocks....Pages 83-105
Principal Blocks....Pages 107-111
General Blocks....Pages 113-138
Off-diagonal Blocks....Pages 139-142
Connections with Linear Systems....Pages 143-176
Applications to Matrix Polynomials....Pages 177-188
Applications to Rational Matrix Functions....Pages 189-216
Infinite Dimensional Operator Blocks....Pages 217-242
Factorization of Operator Polynomials....Pages 243-266
Factorization of Analytic Operator Functions....Pages 267-276
Eigenvalue Completion Problems for Triangular Matrices....Pages 277-295
Back Matter....Pages 297-336

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This book explores a new direction in linear algebra and operator theory dealing 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 problem of classification of partially specified matrices, and the results here may be seen as a far reaching generalization of the Jordan canonical form. The second problem is the eigenvalue completion problem, which asks for a description of 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 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 book will appeal to a wide group of mathematicians and engineers. Much of the material can be used in advanced courses in matrix and operator theory.
Content:
Front Matter....Pages i-viii
Introduction....Pages 1-3
Main Problems and Motivation....Pages 5-23
Elementary Operations on Blocks....Pages 25-38
Full Length Blocks....Pages 39-57
The Eigenvalue Completion Problem for Full Length Blocks....Pages 59-82
Full Width Blocks....Pages 83-105
Principal Blocks....Pages 107-111
General Blocks....Pages 113-138
Off-diagonal Blocks....Pages 139-142
Connections with Linear Systems....Pages 143-176
Applications to Matrix Polynomials....Pages 177-188
Applications to Rational Matrix Functions....Pages 189-216
Infinite Dimensional Operator Blocks....Pages 217-242
Factorization of Operator Polynomials....Pages 243-266
Factorization of Analytic Operator Functions....Pages 267-276
Eigenvalue Completion Problems for Triangular Matrices....Pages 277-295
Back Matter....Pages 297-336
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