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

TL;DR

This paper establishes necessary and sufficient conditions for the existence of rational matrices with prescribed invariant rational functions, minimal indices, and eigenstructure, extending previous results to include column and row space properties.

Contribution

It provides a comprehensive solution to constructing rational matrices with specified eigenstructure and minimal indices, including the less-studied column and row space data.

Findings

Conditions for existence of rational matrices with prescribed eigenstructure and minimal indices.

Extension of previous results to include column and row space minimal indices.

Solutions for polynomial matrices as a key step in the general case.

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.