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 $Q$, $M$, and $R$ where $Q$ and $M$ are self-adjoint. Then the following statements are equivalent:

1. $Q>0$ and $M-R{Q}^{-1}{R}^{*}>0$ both hold.
2. $M>0$ and $Q-R^{*}{M}^{-1}R>0$ both hold.
3. $\left[\begin{array}{cc}M& R\\ R^{*}& Q\end{array}\right]>0$ is satisfied.

More concisely:

$\left[\begin{array}{cc}M& R\\ R^{*}& Q\end{array}\right]>0⟺\left[\begin{array}{cc}M& 0\\ 0& Q-R^{*}{M}^{-1}R\end{array}\right]>0⟺\left[\begin{array}{cc}M-R{Q}^{-1}{R}^{*}& 0\\ 0& Q\end{array}\right]>0$

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.