LMIs in Control/pages/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 ${\displaystyle Q}$, ${\displaystyle M}$, and ${\displaystyle R}$ where ${\displaystyle Q}$ and ${\displaystyle M}$ are self-adjoint. Then the following statements are equivalent:

1. ${\displaystyle Q>0}$ and ${\displaystyle M-RQ^{-1}R^{*}>0}$ both hold.
2. ${\displaystyle M>0}$ and ${\displaystyle Q-R^{*}M^{-1}R>0}$ both hold.
3. ${\displaystyle {\begin{bmatrix}M&R\\R^{*}&Q\end{bmatrix}}>0}$ is satisfied.

More concisely:

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