On a Class of Matrix Pencils and ℓ-ifications Equivalent to a Given Matrix Polynomial

2014/06/04 by Bini, Dario A., Robol, Leonardo
#15A03 #15A21 #65F15 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1406.1025

Abstract

A new class of linearizations and ℓ-ifications for m× m matrix polynomials P(x) of degree n is proposed. The ℓ-ifications in this class have the form A(x) = D(x) + (e⊗ I_m) W(x) where D is a block diagonal matrix polynomial with blocks B_i(x) of size m, W is an m× qm matrix polynomial and e=(1,…,1)^t∈\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 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

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