Factorization of Hermitian matrix polynomials with constant signature (Q1318218)

A new proof is given for the following result proved in the paper by \textit{I. Gohberg}, \textit{P. Lancaster} and \textit{L. Rodman} [Linear Multilinear Algebra 11, 209-224 (1982; Zbl 0486.15008)]. If a polynomial \(A(t)\) with complex Hermitian matrices as coefficients has nonzero determinant and is such that \(A(t)\) has constant signature for all real \(t\) for which \(\text{det} A(t) \neq 0\), then \(A(t)\) admits a factorization \(A(t)=B(t)^* DB(t)\), where \(B(t)\) is a complex matrix polynomial and \(D\) is a nonsingular Hermitian matrix. This theorem is also extended to Laurent polynomial matrices with a suitably defined involution.