  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.
