  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.
