Real Factorization of Positive Semidefinite Matrix Polynomials

TL;DR

This paper characterizes when a real symmetric positive semidefinite matrix polynomial can be factored into a product of a matrix polynomial and its transpose, providing a constructive proof and an explicit algorithm.

Contribution

It offers a necessary and sufficient condition for factorization based on the determinant and presents a constructive proof with an algorithm for computation.

Findings

Factorization exists if and only if the determinant is a perfect square.

Provides a constructive proof using a skew-symmetric solution to a Riccati equation.

Includes a detailed algorithm for computing the factorization.

Abstract

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.