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Polynomial and rational matrices with the invariant rational functions and the four sequences of minimal indices prescribed

Itziar Baragaña, Froilán M. Dopico, Silvia Marcaida, Alicia Roca

TL;DR

This work develops complete existence criteria for polynomial and rational matrices when the column- and row-space minimal indices are prescribed alongside the complete eigenstructure. The authors first solve the polynomial-m matrix inverse problems, employing factorizations, the Smith–McMillan framework, and an index-sum constraint to derive a majorization-type condition that ties degree, invariant factors, and infinity data to the prescribed row/column-space indices. They then extend these results to rational matrices via a least common denominator transformation, supported by a key lemma that preserves subspace data under scaling, yielding analogous necessary and sufficient conditions in the rational case. A central contribution is a reformulation that reduces (P2) to constructing a diagonal-scaled middle factor $E(s)$ so that a structured product $F(s)$ attains the desired degree and infinity multiplicities, enabling constructive solutions for all three problems. The results illuminate how column- and row-space minimal indices interact with invariant factors and infinity data, advancing inverse-problem theory for polynomial and rational matrices and suggesting directions for structured-matrix extensions and open questions.

Abstract

The complete eigenstructure, or structural data, of a rational matrix $R(s)$ is comprised by its invariant rational functions, both finite and at infinity, which in turn determine its finite and infinite pole and zero structures, respectively, and by the minimal indices of its left and right null spaces. These quantities arise in many applications and have been thoroughly studied in numerous references. However, other two fundamental subspaces of $R(s)$ in contrast have received much less attention: the column and row spaces, which also have their associated minimal indices. This work solves the problems of finding necessary and sufficient conditions for the existence of rational matrices in two scenarios: (a) when the invariant rational functions and the minimal indices of the column and row spaces are prescribed, and (b) when the complete eigenstructure together with the minimal indices of the column and row spaces are prescribed. The particular, but extremely important, cases of these problems for polynomial matrices are solved first and are the main tool for solving the general problems. The results in this work complete and non-trivially extend the necessary and sufficient conditions recently obtained for the existence of polynomial and rational matrices when only the complete eigenstructure is prescribed.

Polynomial and rational matrices with the invariant rational functions and the four sequences of minimal indices prescribed

TL;DR

This work develops complete existence criteria for polynomial and rational matrices when the column- and row-space minimal indices are prescribed alongside the complete eigenstructure. The authors first solve the polynomial-m matrix inverse problems, employing factorizations, the Smith–McMillan framework, and an index-sum constraint to derive a majorization-type condition that ties degree, invariant factors, and infinity data to the prescribed row/column-space indices. They then extend these results to rational matrices via a least common denominator transformation, supported by a key lemma that preserves subspace data under scaling, yielding analogous necessary and sufficient conditions in the rational case. A central contribution is a reformulation that reduces (P2) to constructing a diagonal-scaled middle factor so that a structured product attains the desired degree and infinity multiplicities, enabling constructive solutions for all three problems. The results illuminate how column- and row-space minimal indices interact with invariant factors and infinity data, advancing inverse-problem theory for polynomial and rational matrices and suggesting directions for structured-matrix extensions and open questions.

Abstract

The complete eigenstructure, or structural data, of a rational matrix is comprised by its invariant rational functions, both finite and at infinity, which in turn determine its finite and infinite pole and zero structures, respectively, and by the minimal indices of its left and right null spaces. These quantities arise in many applications and have been thoroughly studied in numerous references. However, other two fundamental subspaces of in contrast have received much less attention: the column and row spaces, which also have their associated minimal indices. This work solves the problems of finding necessary and sufficient conditions for the existence of rational matrices in two scenarios: (a) when the invariant rational functions and the minimal indices of the column and row spaces are prescribed, and (b) when the complete eigenstructure together with the minimal indices of the column and row spaces are prescribed. The particular, but extremely important, cases of these problems for polynomial matrices are solved first and are the main tool for solving the general problems. The results in this work complete and non-trivially extend the necessary and sufficient conditions recently obtained for the existence of polynomial and rational matrices when only the complete eigenstructure is prescribed.
Paper Structure (16 sections, 23 theorems, 72 equations)

This paper contains 16 sections, 23 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equivalence relations on rational matrices
  4. Rational subspaces and minimal indices associated to a rational matrix
  5. Polynomial and rational matrices with prescribed eigenstructure
  6. Properties of minimal bases and indices
  7. Statement of the problems for polynomial matrices
  8. Factorizations of polynomial matrices and a reformulation of Problem \ref{['probkl2']}
  9. Solution to the problems for polynomial matrices
  10. Proof of Theorem \ref{['theomainlk2']}: necessity of the conditions
  11. Proof of Theorem \ref{['theomainlk2']}: sufficiency of the conditions
  12. Sufficiency when the prescribed partial multiplicities of $\infty$ are equal to $0$
  13. Sufficiency in the general case
  14. Results for rational matrices
  15. Conclusions and open problems
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $m$, $n$, $r\leq \min\{m,n \}$ be positive integers and $d$ a non negative integer. Let $\alpha_1(s)\mid \dots \mid \alpha_r(s)$ be monic polynomials in $\mathbb F[s]$. Let $(f_r, \ldots, f_{1})$, $(d_1, \ldots, d_{n-r})$, $(v_1, \ldots, v_{m-r})$ be partitions of non negative integers. Then, th

Theorems & Definitions (32)

  • Theorem 2.1: AmBaMaRo23, DeDoVa15 for infinite fields
  • Theorem 2.2: Fo75
  • Remark 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Remark 2.6
  • Definition 2.7: DeDoMaVa16
  • Theorem 2.8: DmDoVa23
  • Corollary 2.9
  • Corollary 2.10
  • ...and 22 more