An elementary and constructive solution to Hilbert's 17th Problem for matrices

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 $\mathbb{R}[x_1,\ldots,x_m]$. The result is that if $A$ is positive semidefinite for all substitutions $(x_1,\ldots,x_m) \in \mathbb{R}^m$, then $A$ can be expressed as a sum of squares of symmetric matrices with entries in $\mathbb{R}(x_1,\ldots,x_m)$. Moreover, our proof is constructive and gives explicit representations modulo the scalar case.