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

摘要

A new class of linearizations and l-ifications for m x m matrix polynomials P(x) of degree n is proposed. The l-ifications in this class have the form A(x) = D(x) + (e circle times I-m)W(x) where D is a block diagonal matrix polynomial with blocks B-i(x) of size m, W is an m x qm matrix polynomial and e = (1, ... ,1)(t) is an element of 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 l-ification, i.e., the reversed polynomial of A(x) defined by A(#)(x) := x(deg A(x))A(x(-1)) is an l-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 l-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. (C) 2015 Elsevier Inc. All rights reserved.