User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/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{align}

Z + XTY + YTX>0, \end{align}$$


 * 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^TPX & F-X^TG_2&H-X^TG_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{align}

X^T-W>0, \end{align}$$


 * 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 $$