Positive Semidefinite Univariate Matrix Polynomials

2017/07/26 by Hanselka, Christoph, Sinn, Rainer
#11E08 #11E25 #13J30 (Secondary) #14P05 (Primary) #47A68 #Algebraic Geometry (math.AG) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1707.08261

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× n can be written as a sum of squares M=Q^TQ, where Q has size (n+1)× 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 Fejér-Riesz Theorem.

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