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

Nonstrict 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) $$\begin{bmatrix}A &B\\ B^T& C \end{bmatrix} \leq 0$$

b) $$A-BC^{+}B^T <0,C \leq 0, B(\mathbf{1}-CC^+)=\mathbf{0},$$ where $$C^+$$ is the Moore-Penrose inverse of $$C.$$

c) $$C-B^TA^{+}B<0,A \leq 0,B^T(\mathbf{1}-AA^+)=\mathbf{0}$$ where $$A^+$$ is the Moore-Penrose inverse of $$A.$$