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

Given a matrix polynomial P(λ)=∑i=0kλiAi of degree k  , where AiAi are n×nn×n matrices with entries in a field FF, the development of linearizations of P(λ)P(λ) that preserve whatever structure P(λ)P(λ) might posses has been a very active area of research in the last decade. Most of the structure-preserving linearizations of P(λ)P(λ) discovered so far are based on certain modifications of block-symmetric linearizations. The block-symmetric linearizations of P(λ)P(λ) 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)DL(P), such that most of its pencils are linearizations. One drawback of the pencils in DL(P)DL(P) is that none of them is a linearization when P(λ)P(λ) is singular. In this paper we introduce new vector spaces of block-symmetric pencils, most of which are strong linearizations of P(λ)P(λ). The dimensions of these spaces are O(n2)O(n2), which, for n≥k, are much larger than the dimension of DL(P)DL(P). When k   is odd, many of these vector spaces contain linearizations also when P(λ)P(λ) is singular. The coefficients of the block-symmetric pencils in these new spaces can be easily constructed as k×kk×k block-matrices whose n×nn×n blocks are of the form 0, ±αIn±αIn, ±αAi±αAi, or arbitrary n×nn×n matrices, where α is an arbitrary nonzero scalar.