Factorization of Matrix and Operator Functions

The State Space Method

By: Harm Bart, Israel Gohberg, Marinus A. Kaashoek

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Hardcover | 19 October 2007

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428 Pages

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22.86 x 16.51 x 2.54

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Thepresentbookdealswithvarioustypesoffactorizationproblemsformatrixand operator functions. The problems appear in di?erent areasof mathematics and its applications. A uni?ed approach to treat them is developed. The main theorems yield explicit necessaryand su?cient conditions for the factorizations to exist and explicit formulas for the corresponding factors. Stability of the factors relative to a small perturbation of the original function is also studied in this book. The unifying theory developed in the book is based on a geometric approach which has its origins in di?erent ?elds. A number of initial steps can be found in: (1) the theory of non-selfadjoint operators, where the study of invariant s- spaces of an operator is related to factorization of the characteristic matrix or operator function of the operator involved, (2) mathematical systems theory and electrical network theory, where a cascade decomposition of an input-output system or a network is related to a fact- ization of the associated transfer function, and (3) thefactorizationtheoryofmatrixpolynomialsintermsofinvariantsubspaces of a corresponding linearization. In all three cases a state space representation of the function to be factored is used, and the factors are expressed in state space form too. We call this approach the state space method. It hasa largenumber of applications.For instance, besides the areasreferred to above, Wiener-Hopf factorizations of some classes of symbols can also be treated by the state space method.

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Motivating Problems, Systems and Realizations
Motivating Problems
Operator Nodes, Systems, and Operations on Systems
Various Classes of Systems
Realization and Linearization of Operator Functions
Factorization and Riccati Equations
Canonical Factorization and Applications
Minimal Realization and Minimal Factorization
Minimal Systems
Minimal Realizations and Pole-Zero Structure
Minimal Factorization of Rational Matrix Functions
Degree One Factors, Companion Based Rational Matrix Functions, and Job Scheduling
Factorization into Degree One Factors
Complete Factorization of Companion Based Matrix Functions
Quasicomplete Factorization and Job Scheduling
Stability of Factorization and of Invariant Subspaces
Stability of Spectral Divisors
Stability of Divisors
Factorization of Real Matrix Functions
Bibliography	p. 393
List of Symbols	p. 401
Index	p. 405
Table of Contents provided by Blackwell. All Rights Reserved.

Factorization of Matrix and Operator Functions

The State Space Method

At a Glance

Hardcover

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How to return your order

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