A State Space Approach to Canonical Factorization with Applications

  • Harm Bart
  • Marinus A. Kaashoek
  • André C. M. Ran
Book

Part of the Operator Theory: Advances and Applications book series (OT, volume 200)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Introduction

    1. Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 1-6
  3. Convolution equations, canonical factorization and the state space method

    1. Front Matter
      Pages 7-7
    2. Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 9-17
    3. Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 19-33
  4. Convolution equations with rational matrix symbols

    1. Front Matter
      Pages 35-35
    2. Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 37-56
    3. Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 57-74
  5. Equations with non-rational symbols

    1. Front Matter
      Pages 75-75
    2. Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 77-113
    3. Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 115-142
    4. Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 143-167
  6. Factorization of selfadjoint rational matrix functions

    1. Front Matter
      Pages 169-169
    2. Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 171-179
    3. Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 181-196
    4. Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 197-209
    5. Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 211-216
  7. Riccati equations and factorization

    1. Front Matter
      Pages 217-217
    2. Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 219-231
    3. Harm Bart, Marinus A. Kaashoek, André C. M. Ran
      Pages 233-247

About this book

Introduction

The present book deals with canonical factorization problems for di?erent classes of matrix and operator functions. Such problems appear in various areas of ma- ematics and its applications. The functions we consider havein common that they appear in the state space form or can be represented in such a form. The main results are all expressed in terms of the matrices or operators appearing in the state space representation. This includes necessary and su?cient conditions for canonical factorizations to exist and explicit formulas for the corresponding f- tors. Also, in the applications the entries in the state space representation play a crucial role. Thetheorydevelopedinthebookisbasedonageometricapproachwhichhas its origins in di?erent ?elds. One of the initial steps can be found in mathematical systems theory and electrical network theory, where a cascade decomposition of an input-output system or a network is related to a factorization of the associated transfer function. Canonical factorization has a long and interesting history which starts in the theory of convolution equations. Solving Wiener-Hopf integral equations is closely related to canonical factorization. The problem of canonical factorization also appears in other branches of applied analysis and in mathematical systems theory, in H -control theory in particular.

Keywords

Matrix algebra convolution factorization matrix function state space

Authors and affiliations

  • Harm Bart
    • 1
  • Marinus A. Kaashoek
    • 2
  • André C. M. Ran
    • 2
  1. 1.Econometrisch InstituutErasumus Universiteit RotterdamRotterdamThe Netherlands
  2. 2.Department of Mathematics, FEWVrije UniversiteitAmsterdamThe Netherlands

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-7643-8753-2
  • Copyright Information Birkhäuser Basel 2010
  • Publisher Name Birkhäuser Basel
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-7643-8752-5
  • Online ISBN 978-3-7643-8753-2
  • About this book