2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/117937We 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.3 pages, generalized and added 2 examplesRings and AlgebrasAlgebraic Geometry12D15, 03C64, 13L05, 14P05, 15A21, 15A54text