A characterization of symmetric positive definite matrices using Schur complements

Question

I have the matrix $$X = \begin{pmatrix} A & B\\ B^\intercal & C\end{pmatrix}\in \mathbb R^{nxn}$$ and $S:= C-B^\intercal A^{-1}B$. by considering $$\min_u \quad u^\intercal Au + 2v^\intercal B^\intercal u+v^\intercal Cv$$ I need to prove that:

1. $X\succ 0$ iff $A\succ 0$ and $S\succ 0$
2. if $A\succ 0$ then $X\succeq 0$ iff $S\succeq 0$

Answer 1

Hint:

$$u^\intercal Au + 2v^\intercal B^\intercal u + v^\intercal Cv = \left(u - A^{-1}B^\intercal v\right)^\intercal A \left(u - A^{-1}B^\intercal v\right) + v^\intercal Sv$$