Block Full Rank Linearizations of Rational Matrices

TL;DR

This paper develops a new family of block full rank linearizations for rational matrices that can recover local information about both zeros and poles, generalizing existing methods and including structures from recent literature.

Contribution

It extends the structure of block full rank pencils to also recover pole information and introduces a new family of linearizations that encompass most existing structures.

Findings

Allows recovery of zeros and poles under minimality conditions

Generalizes existing linearization structures for rational matrices

Includes recent linearizations in the literature

Abstract

Block full rank pencils introduced in [Dopico et al., Local linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems, Linear Algebra Appl., 2020] allow us to obtain local information about zeros that are not poles of rational matrices. In this paper we extend the structure of those block full rank pencils to construct linearizations of rational matrices that allow us to recover locally not only information about zeros but also about poles, whenever certain minimality conditions are satisfied. In addition, the notion of degree of a rational matrix will be used to determine the grade of the new block full rank linearizations as linearizations at infinity. This new family of linearizations is important as it generalizes and includes the structures appearing in most of the linearizations for rational matrices constructed in the literature. In particular, this theory will be applied to study the structure and the properties of the linearizations in [P. Lietaert et al., Automatic rational approximation and linearization of nonlinear eigenvalue problems, submitted].