Schur complement and block positive semidefiniteness

ProofAdvanced⏱ 30 minFoundationspsdschur complementblock matrix

Problem

Let $A \in \mathbb{S}^n$ with $A \succ 0$ , $B \in \R^{n\times m}$ and $C \in \mathbb{S}^m$ . Prove that

M = \begin{pmatrix} A & B \\ B^\top & C\end{pmatrix} \succeq 0 \qquad\Longleftrightarrow\qquad C - B^\top A^{-1} B \succeq 0 .

The matrix $S = C - B^\top A^{-1}B$ is the Schur complement of $A$ in $M$ .

Write a complete argument. Every inequality you use should be one you could state precisely.

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