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An elementary and constructive solution to Hilbert’s 17th problem for matrices. (English) Zbl 1126.12001

A real matrix $A$ is called positive semidefinite if it is symmetric with all nonnegative eigenvalues. The main aim of the paper is the following theorem: If a symmetric matrix $A$ with entries in $R\left[{x}_{1},\cdots ,{x}_{n}\right]$ is positive semidefinite for all sustitutions $\left({x}_{1},\cdots ,{x}_{n}\right)\in {ℝ}^{n}$, then $A$ can be expressed as a sum of squares of symmetric matrices with entries in $R\left({x}_{1},\cdots ,{x}_{n}\right)·$

This generalizes the famous Artin’s theorem solving Hilbert’s 17th problem. The theorem proved by the authors was originally proved by C. Procesi and M. Schacher [Ann. Math. (2) 104, 395–406 (1976; Zbl 0347.16010)]. However, the proof in the paper under review is constructive modulo the fact that one can represent the entries of $A$ as a sum of rational functions.

##### MSC:
 12D15 Formally real fields 13J30 Real algebra 15A21 Canonical forms, reductions, classification 15A54 Matrices over function rings