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*}