# LMIs in Control/Matrix and LMI Properties and Tools/Young’s Relation-Based Properties

## Young’s Relation-Based Properties

1. Consider $X$ ,$Y\in \mathbb {R} ^{n\times m}$ and $Z\in \mathbb {S} ^{m}$ . The matrix inequality given by

{\begin{aligned}Z+XTY+YTX>0,\end{aligned}} is satisfied if and only if there exist $Q\in \mathbb {S} ^{m}$ , $P\in \mathbb {S} ^{n}$ ,$G1\in \mathbb {R} ^{n\times n}$ ,$G2\in \mathbb {R} ^{n\times m}$ ,$F\in \mathbb {R} ^{m\times n}$ , and $H\in \mathbb {R} ^{m\times m}$ , where $Q>0$ and $P>0$ , such that
${\begin{bmatrix}P&Y\\*&Q\end{bmatrix}}>0$ and ${\begin{bmatrix}Z+Q+X^{T}PX&F-X^{T}G_{2}&H-X^{T}G_{1}\\*&G_{1}+G_{1}^{T}-P&F^{T}+G_{2}-Y\\*&*&H^{T}+H-Q\end{bmatrix}}$ 2. Consider $X$ ,$Y\in \mathbb {R} ^{n\times n}$ and $W\in \mathbb {S} ^{m}$ , where $X$ is full rank and $W>0$ . The matrix inequality given by

{\begin{aligned}X^{T}-W>0,\end{aligned}} is satisfied if there exist $\lambda \in \mathbb {R} _{>0}$ such that
${\begin{bmatrix}\lambda \mathbf {1} &\lambda \mathbf {1} &\mathbf {0} \\*&\mathbf {X} +\mathbf {X} ^{T}&\mathbf {W} ^{\frac {1}{2}}\\*&*&\lambda \mathbf {1} \end{bmatrix}}>0$ 