LMIs in Control/Matrix and LMI Properties and Tools/Non 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.

Non Strict Projection Lemma
Consider $$\Psi\in \mathbb{S}^n$$, $$ G\in \mathbb{R}^{n\times m}$$, $$\Lambda\in \mathbb{R}^{m\times p}$$, and $$ H\in \mathbb{R}^{n\times p}.$$, where $$ \mathcal{R} (G)$$ and $$ \mathcal{R} (H)$$ is linear;y independent. There exists $$\Lambda $$ such that:

$$ \begin{align} \ \Psi + G \Lambda H^T + H \Lambda^T G^Tleq0, \end{align} $$

if and only if,

$$ \begin{align} \ N_G^T \Psi N_G \leq0 \end{align} $$,

$$ \begin{align} \ N_H^T \Psi N_H \leq0 \end{align} $$,

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