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arXiv · 0907.2260

Pure states, positive matrix polynomials and sums of hermitian squares

Igor Klep·Markus Schweighofer
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Abstract

Let M be an archimedean quadratic module of real t-by-t matrix polynomials in n variables, and let S be the set of all real n-tuples where each element of M is positive semidefinite. Our key finding is a natural bijection between the set of pure states of M and the cartesian product of S with the real projective (t-1)-space. This leads us to conceptual proofs of positivity certificates for matrix polynomials, including the recent seminal result of Hol and Scherer: If a symmetric matrix polynomial is positive definite on S, then it belongs to M. We also discuss what happens for non-symmetric matrix polynomials or in the absence of the archimedean assumption, and review some of the related classical results. The methods employed are both algebraic and functional analytic.

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Igor Klep, Markus Schweighofer. 2009-09-29. Pure states, positive matrix polynomials and sums of hermitian squares. https://doi.org/10.1512/iumj.2010.59.4107

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