LMIs in Control/pages/Reciprocal Projection Lemma

WIP, Description in progress

Another condition for eliminating a variable in an LMI called reciprocal projection lemma is introduced.

Reciprocal Projection Lemma

For a given symmetric matricx ${\displaystyle \Phi \in \mathbb {S} ^{n}}$ , there exists a matrix ${\displaystyle S\in \mathbb {R} ^{n\times n}}$  satisfying

${\displaystyle \Phi +S^{T}+S<0}$

if and only if, for an arbitrarily fixed symmetric matrix ${\displaystyle P\in \mathbb {S} ^{n}}$ , there exist a matrix ${\displaystyle W\in \mathbb {R} ^{n\times n}}$  satisfying

${\displaystyle {\begin{bmatrix}\Phi +P-(W^{T}+W)&S^{T}+W^{T}\\S+W&-P\end{bmatrix}}<0}$ .