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 $L( \lambda )$.