Hilbert’s 17th problem concerns expression of polynomials on $R^n$ as a sum of squares. It is well known tha tmany positive polynomials are not sums of squares; see [Re], [D’A] for excellent surveys. In thsi paper we consider symmetric noncommutative polynomials and call one “matrix-positive”, if whenever matrices of any size are substituted for the variables in the polynomial the matrix value which the polynomial takes is positive semidefinite. The result in this paper is:
A polynomial is matrix-positive if and only if it is a sum of squares.