On a class of matrix pencils and ℓ-ifications equivalent to a given matrix polynomial

A new class of linearizations and $ell$-ifications for m×m matrix polynomials $P(x)$ of degree $n$ is proposed. The $ell$-ifications in this class have the form $A(x)=D(x)+(eotimes I_m)W(x)$ where $D$ is a block diagonal matrix polynomial with blocks $B_i(x)$ of size $m$, $W$ is an $m imes qm$ matrix polynomial and $e=(1,…,1)^t in C^q$, 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 $A^# (x):=x^{deg ⁡A(x)}A(x^{-1})$ 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.