Linearizations of matrix polynomials viewed as Rosenbrock's system matrices

TL;DR

This paper explores the relationship between classical linearizations of matrix polynomials and Rosenbrock's polynomial system matrices, providing new techniques for identifying linearizations in polynomial eigenvalue problems.

Contribution

It establishes a connection between standard linearization methods and polynomial system matrices, offering novel techniques for recognizing linearizations.

Findings

Connects linearization of matrix polynomials with Rosenbrock's system matrices

Provides new methods to verify if a matrix pencil is a linearization

Enhances tools for solving Polynomial Eigenvalue Problems

Abstract

A well known method to solve the Polynomial Eigenvalue Problem (PEP) is via linearization. That is, transforming the PEP into a generalized linear eigenvalue problem with the same spectral information and solving such linear problem with some of the eigenvalue algorithms available in the literature. Linearizations of matrix polynomials are usually defined using unimodular transformations. In this paper we establish a connection between the standard definition of linearization for matrix polynomials introduced by Gohberg, Lancaster and Rodman and the notion of polynomial system matrix introduced by Rosenbrock. This connection gives new techniques to show that a matrix pencil is a linearization of the corresponding matrix polynomial arising in a PEP.