Linearization of matrix polynomials expressed in polynomial bases

Abstract

This paper concerns regular matrix polynomials P(λ) when represented in various polynomial bases (other than the monomials 1, λ, λ2, …). As in the monomial case, matrices of ‘companion’ form play an important part in theory and numerical practice. In particular, they are used here to construct ‘strong linearizations’ of P(λ). The paper contains three theorems concerning linearizations constructed for representations in a general class of ‘degree-graded’ polynomials, Bernstein polynomials and Lagrange polynomials.