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MiS Preprint

Positive Semidefinite Univariate Matrix Polynomials

Christoph Hanselka and Rainer Sinn


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.

Jul 27, 2017
Aug 7, 2017
MSC Codes:
14P05, 47A68, 11E08, 11E25, 13J30
matrix factorizations, matrix polynomial, sum of squares, Smith normal form

Related publications

2019 Repository Open Access
Christoph Hanselka and Rainer Sinn

Positive semidefinite univariate matrix polynomials

In: Mathematische Zeitschrift, 292 (2019) 1-2, pp. 83-101