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

Abstract Given a matrix polynomial P ( λ ) = ∑ i = 0 k λ i A i of degree k , where A i are n × n matrices with entries in a field F , the development of linearizations of P ( λ ) that preserve whatever structure 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 ) discovered so far are based on certain modifications of block-symmetric linearizations. The block-symmetric linearizations of 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 ) , 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 ( λ ) is singular. In this paper we introduce new vector spaces of block-symmetric pencils, most of which are strong linearizations of P ( λ ) . The dimensions of these spaces are O ( n 2 ) , which, for n ≥ k , are much larger than the dimension of DL ( P ) . When k is odd, many of these vector spaces contain linearizations also when P ( λ ) is singular. The coefficients of the block-symmetric pencils in these new spaces can be easily constructed as k × k block-matrices whose n × n blocks are of the form 0, ± α I n , ± α A i , or arbitrary n × n matrices, where α is an arbitrary nonzero scalar.