Recovering a perturbation of a matrix polynomial from a perturbation of its linearization

TL;DR

This paper develops an algorithm to relate perturbations of a matrix polynomial to perturbations of its linearization, enabling better understanding and control of perturbations in computational problems involving matrix polynomials.

Contribution

It introduces a novel algorithm that maps linearization perturbations back to polynomial coefficient perturbations and finds transformation matrices for strict equivalence.

Findings

Algorithm effectively recovers polynomial perturbations from linearization perturbations

Transforms linearization perturbations into polynomial coefficient perturbations

Applicable to first companion linearization and generalizable to others

Abstract

A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore a perturbation theory results for linearizations need to be related back to matrix polynomials. In this paper we present an algorithm that finds which perturbation of matrix coefficients of a matrix polynomial corresponds to a given perturbation of the entire linearization pencil. Moreover we find transformation matrices that, via strict equivalence, transform a perturbation of the linearization to the linearization of a perturbed polynomial. For simplicity, we present the results for the first companion linearization but they can be generalized to a broader class of linearizations.