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

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 $e=(1,\dots ,1{)}^t\in {\mathbb{C}}^q$, 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

20 pages, 8 figures

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