Pure States, Positive Matrix Polynomials and Sums of Hermitian Squares

Let M be an archimedean quadratic module of real t× t matrix polynomials in n variables, and let S ⊆ R be the set of all points where each element of M is positive semidefinite. Our key finding is a natural bijection between the set of pure states of M and S × P(R). 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 nonsymmetric 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.