LMIs in Control/Matrix and LMI Properties and Tools/Continuous Time/Schur Complement/Strict Schur Complement

Strict Schur Complement
Consider $$A \in \mathbb{S}^n, B \in \mathbb{R}^{n \times m},$$ and $$C \in \mathbb{S}^m.$$ The following statements are equivalent.

a) $$\begin{bmatrix}A &B\\ B^T& C \end{bmatrix}< 0$$

b) $$A-BC^{-1}B^T<0,C<0$$

c) $$C-B^TA^{-1}B<0,A<0$$