User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Schur Complement

An important tool for proving many LMI theorems is the Schur Compliment. It is frequently used as a method of LMI linearization.

The Schur Compliment
Consider the matricies $$Q$$, $$M$$, and $$R$$ where $$Q$$ and $$M$$ are self-adjoint. Then the following statements are equivalent:
 * $$Q>0$$ and $$M-RQ^{-1}R^*>0$$ both hold.
 * $$M>0$$ and $$Q-R^*M^{-1}R>0$$ both hold.
 * $$\begin{bmatrix} M & R \\

R^* & Q\end{bmatrix} > 0 $$ is satisfied.

More concisely:
 * $$\begin{bmatrix} M & R \\ R^* & Q\end{bmatrix} > 0 \iff \begin{bmatrix} M & 0 \\ 0 & Q-R^*M^{-1}R\end{bmatrix} > 0 \iff \begin{bmatrix} M-RQ^{-1}R^* & 0 \\ 0 & Q\end{bmatrix} > 0

$$