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

A new class of linearizations and ℓ  -ifications for m×mm×m matrix polynomials P(x)P(x) of degree n is proposed. The ℓ  -ifications in this class have the form A(x)=D(x)+(e⊗Im)W(x)A(x)=D(x)+(e⊗Im)W(x) where D   is a block diagonal matrix polynomial with blocks Bi(x)Bi(x) of size m, W   is an m×qmm×qm matrix polynomial and e=(1,…,1)t∈Cqe=(1,…,1)t∈Cq, for a suitable integer q  . The blocks Bi(x)Bi(x) can be chosen a priori, subjected to some restrictions. Under additional assumptions on the blocks Bi(x)Bi(x) the matrix polynomial A(x)A(x) is a strong ℓ  -ification, i.e., the reversed polynomial of A(x)A(x) defined by A#(x):=xdeg⁡A(x)A(x−1)A#(x):=xdeg⁡A(x)A(x−1) is an ℓ  -ification of P#(x)P#(x). The eigenvectors of the matrix polynomials P(x)P(x) and A(x)A(x) are related by means of explicit formulas. Some practical examples of ℓ  -ifications are provided. A strategy for choosing Bi(x)Bi(x) in such a way that A(x)A(x) is a well conditioned linearization of P(x)P(x) is proposed. Some numerical experiments that validate the theoretical results are reported.