If the elements of a symmetric matrix lie in the real field it is well known that the roots of its characteristic equation are real. This implies that the discriminant of that equation (i.e. the product of the squared differences of the roots) is a polynomial in the elements which is non-negative and the same must be true for the leading coefficients of all the other Sturm functions associated with the characteristic equation. One would expect that it should be possible to express them as a sum of squares. Conversely, such an expression would establish the reality of the roots.