Symmetric linearizations for matrix polynomials

A standard way of treating the polynomial eigenvalue problem P(λ)x = 0 is to convert it into an equivalent matrix pencil - a process known as linearization. Two vector spaces of pencils double struck L sign1(P) and double struck L sign2(P), and their intersection double struck D sign double struck L sign(P), have recently been defined and studied by Mackey, Mackey, Mehl, and Mehrmann. The aim of our work is to gain new insight into these spaces and the extent to which their constituent pencils inherit structure from P. For arbitrary polynomials we show that every pencil in double struck D sign double struck L sign(P) is block symmetric and we obtain a convenient basis for double struck D sign double struck L sign(P) built from block Hankel matrices. This basis is then exploited to prove that the first deg(P) pencils in a sequence constructed by Lancaster in the 1960s generate double struck D sign double struck L sign(P). When P is symmetric, we show that the symmetric pencils in double struck L sign1(P) comprise double struck D sign double struck L sign(P), while for Hermitian P the Hermitian pencils in double struck L sign1(P) form a proper subset of double struck D sign double struck L sign(P) that we explicitly characterize. Almost all pencils in each of these subsets are shown to be linearizations. In addition to obtaining new results, this work provides a self-contained treatment of some of the key properties of double struck D sign double struck L sign(P) together with some new, more concise proofs. © 2006 Society for Industrial and Applied Mathematics.