Pure states, positive matrix polynomials and sums of hermitian squares

TL;DR

This paper establishes a bijection between pure states of an archimedean quadratic module of matrix polynomials and a product space, providing conceptual proofs for positivity certificates and exploring extensions beyond symmetric polynomials.

Contribution

It introduces a natural bijection linking pure states to a product space, offering new conceptual proofs for positivity certificates of matrix polynomials.

Findings

Bijection between pure states and a product space involving S and projective space

Conceptual proof of Hol and Scherer's positivity result for matrix polynomials

Discussion on non-symmetric polynomials and non-archimedean cases

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.