Strongly minimal self-conjugate linearizations for polynomial and rational matrices

TL;DR

This paper introduces a method to construct strongly minimal linearizations of rational matrices, preserving their structure, which enhances numerical algorithms and generalizes previous approaches for polynomial and rational matrices.

Contribution

It presents a universal approach to create structure-preserving strongly minimal linearizations for any rational matrix, improving upon prior methods that lacked such generality.

Findings

Constructs strongly minimal linearizations from Laurent expansions.

Preserves Hermitian, skew-Hermitian, para-Hermitian, and para-skew-Hermitian structures.

Offers efficient algorithms applicable to all rational matrices.

Abstract

We prove that we can always construct strongly minimal linearizations of an arbitrary rational matrix from its Laurent expansion around the point at infinity, which happens to be the case for polynomial matrices expressed in the monomial basis. If the rational matrix has a particular self-conjugate structure we show how to construct strongly minimal linearizations that preserve it. The structures that are considered are the Hermitian and skew-Hermitian rational matrices with respect to the real line, and the para-Hermitian and para-skew-Hermitian matrices with respect to the imaginary axis. We pay special attention to the construction of strongly minimal linearizations for the particular case of structured polynomial matrices. The proposed constructions lead to efficient numerical algorithms for constructing strongly minimal linearizations. The fact that they are valid for {\em any} rational matrix is an improvement on any other previous approach for constructing other classes of structure preserving linearizations, which are not valid for any structured rational or polynomial matrix. The use of the recent concept of strongly minimal linearization is the key for getting such generality.