Positive semidefinite univariate matrix polynomials
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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 Fejér–Riesz Theorem.
KeywordsMatrix factorizations Matrix polynomial Sum of squares Smith normal form
Mathematics Subject ClassificationPrimary 14P05 Secondary 47A68 11E08 11E25 13J30
We are grateful to Markus Schweighofer. Our approach extends fruitful discussions with him. The first author is supported by the Faculty Research Development Fund (FRDF) of The University of Auckland (Project no. 3709120). The second author would like to thank Bernd Sturmfels and the Max-Planck-Institute in Leipzig for their hospitality and support.
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