Equivalences for Linearizations of Matrix Polynomials

TL;DR

This paper explores new methods for establishing equivalences between matrix polynomials and their linearizations, providing explicit constructions and analyzing their limitations using algebraic and computational techniques.

Contribution

It introduces a novel approach using Hermite Normal Form algorithms to find unimodular cofactors that demonstrate equivalence of matrix polynomials and their linearizations.

Findings

New explicit constructions for linearizations in different polynomial bases

Comparison of unimodular pairs with those from strict equivalence

Discussion on limitations of computational methods for finding unimodular cofactors

Abstract

One useful standard method to compute eigenvalues of matrix polynomials ${\bf P}(z) \in \mathbb{C}^{n\times n}[z]$ of degree at most $\ell$ in $z$ (denoted of grade $\ell$, for short) is to first transform ${\bf P}(z)$ to an equivalent linear matrix polynomial ${\bf L}(z)=z{\bf B}-{\bf A}$, called a companion pencil, where ${\bf A}$ and ${\bf B}$ are usually of larger dimension than ${\bf P}(z)$ but ${\bf L}(z)$ is now only of grade $1$ in $z$. The eigenvalues and eigenvectors of ${\bf L}(z)$ can be computed numerically by, for instance, the QZ algorithm. The eigenvectors of ${\bf P}(z)$, including those for infinite eigenvalues, can also be recovered from eigenvectors of ${\bf L}(z)$ if ${\bf L}(z)$ is what is called a "strong linearization" of ${\bf P}(z)$. In this paper we show how to use algorithms for computing the Hermite Normal Form of a companion matrix for a scalar polynomial to direct the discovery of unimodular matrix polynomial cofactors ${\bf E}(z)$ and ${\bf F}(z)$ which, via the equation ${\bf E}(z){\bf L}(z){\bf F}(z) = \mathrm{diag}( {\bf P}(z), {\bf I}_n, \ldots, {\bf I}_n)$, explicitly show the equivalence of ${\bf P}(z)$ and ${\bf L}(z)$. By this method we give new explicit constructions for several linearizations using different polynomial bases. We contrast these new unimodular pairs with those constructed by strict equivalence, some of which are also new to this paper. We discuss the limitations of this experimental, computational discovery method of finding unimodular cofactors.