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Rosenbrock's Theorem on System Matrices over Elementary Divisor Domains

Froilán Dopico, Vanni Noferini, Ion Zaballa

Abstract

Rosenbrock's theorem on polynomial system matrices is a classical result in linear systems theory that relates the Smith-McMillan form of a rational matrix $G$ with the Smith forms of an irreducible polynomial system matrix $P$ giving rise to $G$ and of a submatrix of $P$. This theorem has been essential in the development of algorithms for computing the poles and zeros of a rational matrix via linearizations and generalized eigenvalue algorithms. In this paper, we extend Rosenbrock's theorem to system matrices $P$ with entries in an arbitrary elementary divisor domain $\mathfrak R$ and matrices $G$ with entries in the field of fractions of $\mathfrak R$. These are the most general rings where the involved Smith-McMillan and Smith forms both exist and, so, where the problem makes sense. Moreover, we analyze in detail what happens when the system matrix is not irreducible. Finally, we explore how Rosenbrock's theorem can be extended when the system matrix $P$ itself has entries in the field of fractions of the elementary divisor domain.

Rosenbrock's Theorem on System Matrices over Elementary Divisor Domains

Abstract

Rosenbrock's theorem on polynomial system matrices is a classical result in linear systems theory that relates the Smith-McMillan form of a rational matrix with the Smith forms of an irreducible polynomial system matrix giving rise to and of a submatrix of . This theorem has been essential in the development of algorithms for computing the poles and zeros of a rational matrix via linearizations and generalized eigenvalue algorithms. In this paper, we extend Rosenbrock's theorem to system matrices with entries in an arbitrary elementary divisor domain and matrices with entries in the field of fractions of . These are the most general rings where the involved Smith-McMillan and Smith forms both exist and, so, where the problem makes sense. Moreover, we analyze in detail what happens when the system matrix is not irreducible. Finally, we explore how Rosenbrock's theorem can be extended when the system matrix itself has entries in the field of fractions of the elementary divisor domain.

Paper Structure

This paper contains 10 sections, 26 theorems, 103 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background material
  3. Rosenbrock's theorem over EDDs
  4. Beyond Rosenbrock's theorem
  5. A further improvement of Theorem \ref{['thm.mainbeyond']} over PIDs
  6. Three corollaries of Rosenbrock's theorem
  7. Rosenbrock's theorem for matrices over $\mathbb{F}$
  8. Conclusions
  9. Some divisibility properties over Greatest Common Divisor Domains
  10. Proof of Proposition \ref{['teo_copr_an']}

Key Result

Proposition 2.1

Let $G_1\in\mathfrak R^{p\times m}$ and $G_2\in\mathfrak R^{q\times m}$, $p+q\geq m$. The following are equivalent:

Figures (1)

  • Figure 1: Inclusion relations between different classes of commutative rings discussed in this paper. All inclusions in the figure are known to be strict with the exception of the one between Bézout domains and elementary divisor domains

Theorems & Definitions (35)

  • Proposition 2.1
  • Proposition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Remark 3.4
  • Theorem 3.5
  • Proposition 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Remark 4.4
  • ...and 25 more