Singularities of rational functions and minimal factorizations: The noncommutative and the commutative setting

Dmitry S. Kaliuzhnyi-Verbovetskyi, Victor Vinnikov

Research output: Contribution to journalArticlepeer-review

50 Scopus citations

Abstract

We show that the singularities of a matrix-valued noncommutative rational function which is regular at zero coincide with the singularities of the resolvent in its minimal state space realization. The proof uses a new notion of noncommutative backward shifts. As an application, we establish the commutative counterpart of the singularities theorem: the singularities of a matrix-valued commutative rational function which is regular at zero coincide with the singularities of the resolvent in any of its Fornasini-Marchesini realizations with the minimal possible state space dimension. The singularities results imply the absence of zero-pole cancellations in a minimal factorization, both in the noncommutative and in the commutative setting.

Original languageEnglish
Pages (from-to)869-889
Number of pages21
JournalLinear Algebra and Its Applications
Volume430
Issue number4
DOIs
StatePublished - 1 Feb 2009

Keywords

  • Minimal factorization
  • Minimal realization
  • Noncommutative
  • Rational function
  • Singularities

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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