Large vector spaces of block-symmetric strong linearizations of matrix polynomials

Abstract

Given a matrix polynomial P(lambda) = Sigma(k)(i=0) lambda(i) A(i) of degree k, where A(i) are n x n matrices with entries in a field F, the development of linearizations of P(lambda) that preserve whatever structure P(lambda) might posses has been a very active area of research in the last decade. Most of the structure-preserving linearizations of P(lambda) discovered so far are based on certain modifications of block-symmetric linearizations. The block-symmetric linearizations of P(lambda) available in the literature fall essentially into two classes: linearizations based on the so-called Fiedler pencils with repetition, which form a finite family, and a vector space of dimension k of block-symmetric pencils, called DL(P), such that most of its pencils are linearizations. One drawback of the pencils in DL(P) is that none of them is a linearization when P(lambda) is singular. In this paper we introduce new vector spaces of block,symmetric pencils, most of which are strong linearizations of P(lambda). The dimensions of these spaces are O(n(2)), which, for n >= root k, are much larger than the dimension of DL(P). When k is odd, many of these vector spaces contain linearizations also when P(lambda) is singular. The coefficients of the block-symmetric pencils in these new spaces can be easily constructed as k x k block-matrices whose n x n blocks are of the form 0, +/-alpha I-n, +/-alpha A(i), or arbitrary n x n matrices, where a is an arbitrary nonzero scalar.