Bart / Gohberg Factorization of Matrix and Operator Functions: The State Space Method

1. Auflage 2007
ISBN: 978-3-7643-8268-1
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 412 Seiten

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.

Autoren/Hrsg.

Bart, Harm

Gohberg, Israel

Inhaltsverzeichnis

1;Contents;7
2;Preface;11
3;Introduction;13
4;Motivating Problems, Systems and Realizations;16
5;Motivating Problems;17
6;Operator Nodes, Systems, and Operations on Systems;34
7;Various Classes of Systems;58
8;Realization and Linearization of Operator Functions;74
9;Factorization and Riccati Equations;86
10;Canonical Factorization and Applications;98
11;Minimal Realization and Minimal Factorization;111
12;Minimal Systems;112
13;Minimal Realizations and Pole-Zero Structure;136
14;Minimal Factorization of Rational Matrix Functions;169
15;Degree One Factors, Companion Based Rational Matrix Functions, and Job Scheduling;186
16;Factorization into Degree One Factors;187
17;Complete Factorization of Companion Based Matrix Functions;215
18;Quasicomplete Factorization and Job Scheduling;271
19;Stability of Factorization and of Invariant Subspaces;320
20;Stability of Spectral Divisors;321
21;Stability of Divisors;341
22;Factorization of Real Matrix Functions;376
23;Bibliography;393
24;List of Symbols;401
25;Index;405