P3.3 Consider the problem
$\qquad \text{minimize } f(x, y) = \begin{bmatrix} x^T & y^T \end{bmatrix} \underbrace{\begin{bmatrix} A & B \ B^T & C \end{bmatrix}}_P \begin{bmatrix} x \ y \end{bmatrix}$
with variables $x$ and $y$. Assume that $A$ is symmetric and positive definite.
(a) Find the solution with respect to $x$ in terms of $y$. That is fix $y$ and solve for $x$.
(b) Find the optimal value $\min_x f(x, y) = f(x^*, y)$.
(c) Conclude that $P$ is positive definite if and only if $A$ and $S = C - B^T A^{-1} B$ are
positive definite matrices.
