Positive semi-definite matrices as sums of squares

Abstract

If K is a field and char K ≠ 2, then an element α ϵ K is a sum of squares in K if and only if α ⩾ 0 for every ordering of K. This is the famous theorem of Artin and Landau. It has been generalized to symmetric matrices over K by D. Gondard and P. Ribenboim. They have also shown that Artin's theorem on positive definite rational functions has a suitable extension to positive definite matrix functions. In this paper we attain two goals. First, we show that similar theorems are valid for Hermitian matrices instead of symmetric ones. Second, we simplify D. Gondard and P. Ribenboim's proof of their second theorem and strengthen it.