I think it's easier here to describe positive definiteness in terms of inner products rather than eigenvalues. If denotes the inner product in , and is a matrix, then A is positive definite if (and only if) for all x in .

The condition for M to be positive definite is thus (for all x in and all y in ). Now take in that inequality and simplify the result. (You will need the property of an inner product, that .) You should end up with the inequality , which shows that S is positive definite.