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

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

2006-10-12

arxiv.org/abs/math/0610388v3

Mathematics

Rings and Algebras

3 pages, generalized and added 2 examples

Scientific paper

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.

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