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 L1(P ) and L2(P ), and their intersection DL(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 DL(P ) is block symmetric and we obtain a convenient basis for DL(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 DL(P ). When P is symmetric, we show that the symmetric pencils in L1(P ) comprise DL(P ), while for Hermitian P the Hermitian pencils in L1(P ) form a proper subset of DL(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 DL(P ) together with some new, more concise proofs.