Real factorization of positive semidefinite matrix polynomials

Suppose Q(x) is a real n x n regular symmetric positive semidefinite matrix polynomial. Then it can be factored as Q(x) = G(x)TG(x), where G(x) is a real n x n matrix polynomial with degree half that of Q(x) if and only if det(Q(x)) is the square of a nonzero real polynomial. We provide a constructive proof of this fact, rooted in finding a skew-symmetric solution to a modified algebraic Riccati equation XSX- XR +RTX + P = 0, where P, R, S are real n x n matrices with P and S real symmetric. In addition, we provide a detailed algorithm for computing the factorization. (c) 2023 Elsevier Inc. All rights reserved.