User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Strict Projection Lemma

Introduction
Projection Lemma is also known as Matrix Elimination Lemma. Strict Projection Lemma is one of the characteristics of the Projection Lemma.

Strict Projection Lemma
Consider $$\Psi\in \mathbb{S}^n$$, $$ G\in \mathbb{R}^{n\times m}$$, $$\Lambda\in \mathbb{R}^{m\times p}$$, and $$ H\in \mathbb{R}^{n\times p}.$$ There exists $$\Lambda $$ such that:

$$ \begin{align} \ \Psi + G \Lambda H^T + H \Lambda^T G^T<0, \end{align} $$

if and only if,

$$ \begin{align} \ N_G^T \Psi N_G <0 \end{align} $$,

$$ \begin{align} \ N_H^T \Psi N_H <0 \end{align} $$,

where $$\mathcal{R} (N_G) =\mathcal{N} (G^T)$$ and $$\mathcal{R} (N_H) = \mathcal{N} (H^T)$$.