Positive Semidefinite Univariate Matrix Polynomials

TL;DR

This paper provides a new proof for the sum-of-squares representation of positive semidefinite univariate matrix polynomials, establishing a link with sums of two squares and proving a matrix Fejér-Riesz type theorem.

Contribution

It introduces a quadratic form approach to prove minimal sum-of-squares representations and confirms a conjecture relating these to sums of two squares of the determinant.

Findings

Every positive semidefinite univariate matrix polynomial can be expressed as a sum of squares with a specific size.

Minimal sum-of-squares representations are generically in one-to-one correspondence with sums of two squares of the determinant.

Every positive semidefinite hermitian univariate matrix polynomial is a square, confirming the matrix Fejér-Riesz theorem.

Abstract

We study sum-of-squares representations of symmetric univariate real matrix polynomials that are positive semidefinite along the real line. We give a new proof of the fact that every positive semidefinite univariate matrix polynomial of size $n\times n$ can be written as a sum of squares $M=Q^TQ$, where $Q$ has size $(n+1)\times n$, which was recently proved by Blekherman-Plaumann-Sinn-Vinzant. Our new approach using the theory of quadratic forms allows us to prove the conjecture made by these authors that these minimal representations $M=Q^TQ$ are generically in one-to-one correspondence with the representations of the nonnegative univariate polynomial $\det(M)$ as sums of two squares. In parallel, we will use our methods to prove the more elementary hermitian analogue that every hermitian univariate matrix polynomial $M$ that is positive semidefinite along the real line, is a square, which is known as the matrix Fej\'er-Riesz Theorem.