  We give a short and elementary proof of a theorem of Procesi, Schacher and
(independently) Gondard, Ribenboim that generalizes a famous result of Artin.
Let $A$ be an $n \times n$ symmetric matrix with entries in the polynomial ring
$\mathbb R[x_1,...,x_m]$. The result is that if $A$ is postive semidefinite for
all substitutions $(x_1,...,x_m) \in \mathbb R^m$, then $A$ can be expressed as
a sum of squares of symmetric matrices with entries in $\mathbb
R(x_1,...,x_m)$. Moreover, our proof is constructive and gives explicit
representations modulo the scalar case.
