Positive semidefinite univariate matrix polynomials

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 Fejer–Riesz Theorem.