Schur complement: characterizations of positive definiteness for block matrices

Let a block matrix $X\in \mathbf{S}^n$ be partitioned as

$$X = \begin{bmatrix}                  A      &B \\                  B^T  &C            \end{bmatrix}$$ where $A\in \mathbf{S}^k$.  If $\det A \neq 0$, the schur complement is
$$S =C - B^TA^{-1}B$$ The following characterizations of positive definiteness can be derived

• $X\succ 0$ iff $A\succ 0$ and $S\succ 0$
• If $A\succ 0$, then $X\succeq 0$ iff $S\succeq 0$

Example:
$$\begin{align*} &A^TA \preceq t^2I, \quad &t\ge 0 \\ \Longleftrightarrow \quad &tI-t^{-1}A^T I A \succeq 0, \quad &t\ge 0 \\ \Longleftrightarrow \quad &   \begin{bmatrix}      tI & A \\      A^T & tI   \end{bmatrix} \succeq 0 \end{align*}$$