Inversibility of a block matrix in schur decomposition

Question

Let $A'$ be a given, n × n, real, positive definite matrix partitioned as follows: \begin{pmatrix} A & B \\ B^T & C \end{pmatrix}

show that $C − B^TA^{−1}B$ is positive definite. I know that I have to do something like this . But when I started writing the solution, I come up with a question. Can I show that $A$ is invertible or is it a necessary condition?

Answer 1

If $A'$ is symmetric positive definite, its principal submatrix $A$ must also be symmetric positive definite, because $x^TAx=\pmatrix{x^T&0}A'\pmatrix{x\\ 0}>0$ for every nonzero vector $x$. Hence $A$ is necessarily nonsingular.