Strong linearizations of rational matrices with polynomial part expressed in an orthogonal basis

TL;DR

This paper introduces a new family of strong linearizations for rational matrices with polynomial parts expressed in orthogonal bases, enabling better eigenvector recovery and structure preservation.

Contribution

It develops a novel approach combining existing theories to construct strong linearizations for rational matrices in orthogonal polynomial bases, including symmetric and Hermitian cases.

Findings

New strong linearizations for rational matrices are constructed.

Eigenvectors of rational matrices can be recovered from their linearizations.

The methods extend to various polynomial bases and preserve matrix structures.

Abstract

We construct a new family of strong linearizations of rational matrices considering the polynomial part of them expressed in a basis that satisfies a three term recurrence relation. For this purpose, we combine the theory developed by Amparan et al., MIMS EPrint 2016.51, and the new linearizations of polynomial matrices introduced by Fa{\ss}bender and Saltenberger, Linear Algebra Appl., 525 (2017). In addition, we present a detailed study of how to recover eigenvectors of a rational matrix from those of its linearizations in this family. We complete the paper by discussing how to extend the results when the polynomial part is expressed in other bases, and by presenting strong linearizations that preserve the structure of symmetric or Hermitian rational matrices. A conclusion of this work is that the combination of the results in this paper with those in Amparan et al., MIMS EPrint 2016.51, allows us to use essentially all the strong linearizations of polynomial matrices developed in the last fifteen years to construct strong linearizations of any rational matrix by expressing such matrix in terms of its polynomial and strictly proper parts.