On the sign characteristic of Hermitian linearizations in DL(P)

In this paper, for a Hermitian matrix polynomial P(λ) with nonsingular leading coefficient, we describe, in terms of the sign characteristic of P(λ), the sign characteristic of the Hermitian linearizations in the vector space DL(P) (Mackey, Mackey, Mehl and Mehrmann, 2006). In particular, we identify the Hermitian linearizations in DL(P) that preserve the sign characteristic of P(λ). We also provide a description of the sign characteristic of the Hermitian linearizations of P(λ) in the family of generalized Fiedler pencils with repetition (Bueno, Dopico, Furtado and Rychnovsky, 2015).