On the Sign Characteristic of Hermitian Linearizations in Dl(p )

The computation of eigenvalues and eigenvectors of matrix polynomials is an important, but di cult, problem. The standard approach to solve this problem is to use linearizations, which are matrix polynomials of degree 1 that share the eigenvalues of P ( ). Hermitian matrix polynomials and their real eigenvalues are of particular interest in applications. Attached to these eigenvalues is a set of signs called the sign characteristic. From both a theoretical and a practical point of view, it is important to be able to recover the sign characteristic of a Hermitian linearization of P ( ) from the sign characteristic of P ( ). In this paper, for a Hermitian matrix polynomial P ( ) with nonsingular leading coe cient, we describe, in terms of the sign characteristic of P ( ), 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 ( ). We also provide a description of the sign characteristic of the Hermitian linearizations of P ( ) in the family of generalized Fiedler pencils with repetition (Bueno, Dopico, Furtado and Rychnovsky, 2015).