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

{\displaystyle {\begin{aligned}\ \Psi +G\Lambda H^{T}+H\Lambda ^{T}G^{T}leq0,\end{aligned}}}

if and only if,

{\displaystyle {\begin{aligned}\ N_{G}^{T}\Psi N_{G}\leq 0\end{aligned}}} ,

{\displaystyle {\begin{aligned}\ N_{H}^{T}\Psi N_{H}\leq 0\end{aligned}}} ,

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