Algebraic Linearizations of Matrix Polynomials

TL;DR

This paper introduces methods to construct linearizations of complex matrix polynomials by building upon linearizations of their component parts, extending companion matrix techniques.

Contribution

It presents a systematic approach to derive linearizations of various matrix polynomial forms from existing linearizations of their factors.

Findings

Provides explicit constructions for linearizations of combined matrix polynomials.

Extends companion matrix methods to broader classes of matrix polynomials.

Facilitates numerical solutions of polynomial eigenvalue problems.

Abstract

We show how to construct linearizations of matrix polynomials $z\mathbf{a}(z)\mathbf{d}_0 + \mathbf{c}_0$, $\mathbf{a}(z)\mathbf{b}(z)$, $\mathbf{a}(z) + \mathbf{b}(z)$ (when $\mathrm{deg}\left(\mathbf{b}(z)\right) < \mathrm{deg}\left(\mathbf{a}(z)\right)$), and $z\mathbf{a}(z)\mathbf{d}_0\mathbf{b}(z) + \mathbf{c_0}$ from linearizations of the component parts, $\mathbf{a}(z)$ and $\mathbf{b}(z)$. This allows the extension to matrix polynomials of a new companion matrix construction.