Factorization of hermitian matrix polynomials with constant signature

Abstract

Gohberg, Lancaster, and Rodman have shown that if a polynomial A( t), with complex hermitian matrices as coefficients, has nonzero determinant and is such that A(λ) has a constant signature for all real λ for which det A(λ) ≠ 0, then A( t) admits a factorization A(t)= B ∗(t)DB(t) , where B( t) is a polynomial with complex matrices as coefficients and D is a nonsingular complex hermitian matrix (which may be assumed to be diagonal with diagonal entries ±1). We give a new, short, and simple proof of this theorem and extend it to Laurent polynomial matrices with a suitably defined involution.