On self-adjoint matrix polynomials with constant signature

Abstract This paper mainly concerns the class of self-adjoint matrix polynomials with constant signature. We define the order of neutrality for a regular self-adjoint matrix polynomial L(λ). Suppose that the leading coefficient of L(λ) is nonsingular; then it is proved that L(λ) is of constant signature if and only if nl is even and the order of neutrality of L(λ) is equal to nl 2 , where n is the degree l is the size of the polynomial L(λ). A similar result is obtained for regular self-adjoint matrix polynomials. We give an answer to an open problem concerning symmetric factorizations of self-adjoint matrix polynomials of constant signature. Let L(λ) be a self-adjoint matrix polynomial of even degree and constant signature and with a nonsingular leading coefficient An. If all the elementary divisors of L(λ) are linear, then L(λ) = [M( \ gl)]∗A n M(λ) , with M(λ) a monic matrix polynomial. In particular, this shows that generically a self-adjoint matrix polynomial of even degree and constant signature admits such a factorization. The proof of this factorization result is based on a result concerning invariant maximal neutral subspaces for a matrix which is self-adjoint in an indefinite inner product. The latter result also applies to other situations, e.g., to factorizations of rational matrix functions with constant signature and to the existence of hermitian solutions for a class of algebraic Riccati equations.