Real factorization of positive semidefinite matrix polynomials

Abstract

Suppose Q(x) is a real n×n regular symmetric positive semidefinite matrix polynomial. Then it can be factored asQ(x)=G(x)TG(x), where G(x) is a real n×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 equationXSX−XR+RTX+P=0, where P,R,S are real n×n matrices with P and S real symmetric. In addition, we provide a detailed algorithm for computing the factorization.