Let M be a real symmetric and positive definite matrix with blocks A, Bt, B and C.

where is a matrix; is ; and is .

Let be the Schur complement. We prove that S is symmetric positive definite.

I can prove that S is symmetric but I am having trouble proving that it is positive definite.

I know that for S a symmetric matrix, S positive definite is equivalent to say that all eigen values of S are positive.

I guess my question is how do we prove that all eigen values of S are positive?