Positive Semidefinite Univariate Matrix Polynomials

Christoph Hanselka and Rainer Sinn

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Submission date: 27. Jul. 2017
Pages: 20
published in: Mathematische Zeitschrift, 292 (2019) 1-2, p. 83-101 
DOI number (of the published article): 10.1007/s00209-018-2137-7
Bibtex
MSC-Numbers: 14P05, 47A68, 11E08, 11E25, 13J30
Keywords and phrases: matrix factorizations, matrix polynomial, sum of squares, Smith normal form
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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.