Factorization of Matrix Polynomials with Symmetries

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