ON SELF-ADJOINT MATRIX POLYNOMIALS WITH CONSTANT SIGNATURE

This paper mainly concerns the class of self-adjoint matrix polynomial s with constant signature. We define the order of neutrality for a reg ular self-adjoint matrix polynomial L(lambda). Suppose-that the leadin g coefficient of L(lambda) is nonsingular; then it is proved that L(la mbda) is of constant signature if and only if nl is even and the order of neutrality of L(lambda) is equal to nl/2, where n is the degree an d l is the size of the polynomial L(lambda). A similar result is obtai ned for regular self-adjoint matrix polynomials. We give an answer to an open problem concerning symmetric factorizations of self-adjoint ma trix polynomials of constant signature. Let L(lambda) be a self-adjoin t matrix polynomial of even degree and constant signature and with a n onsingular leading coefficient A(n). If all the elementary divisors of L(lambda) are linear, then L(lambda) = [M(<(lambda)over bar>)]A(n)M( lambda), with M(lambda) a monic matrix polynomial. In particular, this shows that generically a self-adjoint matrix polynomial of even degre e and constant signature admits such a factorization. The proof of thi s factorization result is based on a result concerning invariant maxim al neutral subspaces for a matric; which is self-adjoint in an indefin ite inner product. The latter result also applies to other situations, e.g., to factorizations of rational matrix functions with constant si gnature and to the existence of hermitian solutions for a class of alg ebraic Riccati equations. (C) Elsevier Science Int., 1997.