Factorization of Matrix Polynomials with Symmetries

An $n \times n$ matrix polynomial $L(\lambda)$ (with real or complex coefficients) is called self-adjoint if $L(\lambda) = (L(\bar{\lambda}))*$ and symmetric if $L(\lambda) = (L(\pm\lambda))^T$. Factorizations of selfadjoint and symmetric matrix polynomials of the form $L(\lambda) = (M(\bar{\lambda}))*DM(\lambda)$ or $L(\lambda) = (M(\pm\lambda))^{T}DM(\lambda)$ are studied, where $D$ is a constant matrix and $M(\lambda)$ is a matrix polynomial. In particular, the minimal possible size of $D$ is described in terms of the elementary divisors of $L(\lambda)$ and (sometimes) signature of the Hermitian values of $L(\lambda)$.