FACTORIZATION OF HERMITIAN MATRIX POLYNOMIALS WITH CONSTANT SIGNATURE

Gohberg, Lancaster, and Rodman have shown that if a polynomial A(t), w ith complex hermitian matrices as coefficients, has nonzero determinan t and is such that A(lambda) has constant signature for all real lambd a for which det A(lambda) not-equal 0, then A(t) admits a factorizatio n 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.