## 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 language | English |
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Pages (from-to) | 869-889 |

Number of pages | 21 |

Journal | Linear Algebra and Its Applications |

Volume | 430 |

Issue number | 4 |

DOIs | |

State | Published - 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