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Factorization of Matrix and Operator Functions: The State Space Method

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  • © 2008
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Authors:
  1. Harm Bart

    1. Econometrisch Instituut, Erasmus Universiteit Rotterdam, Rotterdam, The Netherlands

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  2. André C. M. Ran

    1. Department of Mathematics, FEW, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

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  3. Israel Gohberg

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

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  4. Marinus A. Kaashoek

    1. Department of Mathematics, FEW, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

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  • Second book which is devoted to the state space factorization theory; the first appeared in 1979 as volume 1 of this book series; it contains a substantial selection from the first book, in a reorganized and updated form
  • An entirely new part is devoted to the theory of factorization into degree one factors and its connection to the combinatorial problem of job scheduling in operations research; it is completely finite dimensional and can be considered as a new advanced chapter of Linear Algebra and its Applications
  • Almost each chapter offers new elements and in many cases new sections, taking into account a number of new results in state space factorization theory and its applications that have appeared in the period of 25 years after publication of the first book
  • Stronger emphasis on non-minimal factorization
  • Includes supplementary material: sn.pub/extras

Part of the book series: Operator Theory: Advances and Applications (OT, volume 178)

Part of the book sub series: Linear Operators and Linear Systems (LOLS)

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Table of contents (16 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    1. Introduction

      Pages 1-3
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  5. Degree One Factors, Companion Based Rational Matrix Functions, and Job Scheduling

    1. Front Matter

      Pages 181-181
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    2. Factorization into Degree One Factors

      Pages 183-210
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    3. Complete Factorization of Companion Based Matrix Functions

      Pages 211-266
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    4. Quasicomplete Factorization and Job Scheduling

      Pages 267-315
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  6. Stability of Factorization and of Invariant Subspaces

    1. Front Matter

      Pages 317-317
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    2. Stability of Spectral Divisors

      Pages 319-338
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    3. Stability of Divisors

      Pages 339-373
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Keywords

About this book

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.

Authors and Affiliations

  • Econometrisch Instituut, Erasmus Universiteit Rotterdam, Rotterdam, The Netherlands

    Harm Bart

  • Department of Mathematics, FEW, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

    André C. M. Ran, Marinus A. Kaashoek

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

    Israel Gohberg

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