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Factorization of Matrix and Operator Functions: The State Space Method

Harm Bart, Israel Gohberg, Marinus A. Kaashoek, André C.M. Ran (Autoren)

Buch | Hardcover

XII, 412 Seiten

2007
Springer Basel (Verlag)
978-3-7643-8267-4 (ISBN)

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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.

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.

Erscheint lt. Verlag	19.10.2007
Reihe/Serie	Linear Operators and Linear Systems
Reihe/Serie	Operator Theory: Advances and Applications
Zusatzinfo	XII, 412 p.
Verlagsort	Basel
Sprache	englisch
Maße	170 x 244 mm
Gewicht	915 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
	Mathematik / Informatik ► Mathematik ► Analysis
	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
Schlagworte	convolution • Factorization • Hardcover, Softcover / Mathematik/Analysis • HC/Mathematik/Analysis • Matrix • matrix function • matrix theory • Matrizen • Operatoren • State space method
ISBN-10	3-7643-8267-8 / 3764382678
ISBN-13	978-3-7643-8267-4 / 9783764382674
Zustand	Neuware