On a Class of Matrix Pencils and $\ell$-ifications Equivalent to a Given Matrix Polynomial

TL;DR

This paper introduces a new class of linearizations and $ ext{ extl{}-ifications}$ for matrix polynomials, providing explicit formulas, practical examples, and strategies for conditioning, supported by numerical experiments.

Contribution

It proposes a novel class of $ ext{ extl{}-ifications}$ with explicit formulas and conditions, enhancing the understanding and construction of linearizations for matrix polynomials.

Findings

Explicit formulas relate eigenvectors of $P(x)$ and $A(x)$.

Conditions under which $A(x)$ is a strong $ ext{ extl{}-ification}$.

Numerical experiments validate theoretical results.

Abstract

A new class of linearizations and $\ell$-ifications for $m\times m$ matrix polynomials $P(x)$ of degree $n$ is proposed. The $\ell$-ifications in this class have the form $A(x) = D(x) + (e\otimes I_m) W(x)$ where $D$ is a block diagonal matrix polynomial with blocks $B_i(x)$ of size $m$, $W$ is an $m\times qm$ matrix polynomial and $e=(1,\ldots,1)^t\in\mathbb C^q$, for a suitable integer $q$. The blocks $B_i(x)$ can be chosen a priori, subjected to some restrictions. Under additional assumptions on the blocks $B_i(x)$ the matrix polynomial $A(x)$ is a strong $\ell$-ification, i.e., the reversed polynomial of $A(x)$ defined by $A^\#(x) := x^{\mathrm{deg} A(x)} A(x^{-1})$ is an $\ell$-ification of $P^\#(x)$. The eigenvectors of the matrix polynomials $P(x)$ and $A(x)$ are related by means of explicit formulas. Some practical examples of $\ell$-ifications are provided. A strategy for choosing $B_i(x)$ in such a way that $A(x)$ is a well conditioned linearization of $P(x)$ is proposed. Some numerical experiments that validate the theoretical results are reported