Real Factorization of Positive Semidefinite Matrix Polynomials

Suppose $Q(x)$ is a real $n\times n$ regular symmetric positive semidefinite matrix polynomial. Then it can be factored as $$Q(x) = G(x)^TG(x),$$ where $G(x)$ is a real $n\times n$ 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 $$XSX - XR + R^TX + P = 0,$$ where $P,R,S$ are real $n\times n$ matrices with $P$ and $S$ real symmetric. In addition, we provide a detailed algorithm for computing the factorization