LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Reciprocal Projection Lemma

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

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

$$\Phi +S^T+ S < 0$$

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

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

WIP, additional references to be added