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

Abstract

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 A be an n × n n \times n symmetric matrix with entries in the polynomial ring R [ x 1 , … , x m ] \mathbb R[x_1,\ldots ,x_m] . The result is that if A A is positive semidefinite for all substitutions ( x 1 , … , x m ) ∈ R m (x_1,\ldots ,x_m) \in \mathbb R^m , then A A can be expressed as a sum of squares of symmetric matrices with entries in R ( x 1 , … , x m ) \mathbb R(x_1,\ldots ,x_m) . Moreover, our proof is constructive and gives explicit representations modulo the scalar case.