Does anyone know the name of this following equation or where it is from?
For any given $x,y \in \mathbb{R}^n$, and matrices $P>0$, $D$ and $S$ of appropriate dimensions, one has \begin{equation} 2x^TDSy \leq x^T DPD^T x + y^T S^TP^{-1}Sy. \end{equation}
I read that in one paper but the author does not give any specific instruction. Then I am supposed to think that should be a regular equation in matrix theory. But I check the book about matrix theory without any finding. Could you help me to identify what it is?
This is a quadratic form of the subtraction as $(x^TD\sqrt{P}-y^TS^T\sqrt{P^{-1}})(x^TD\sqrt{P}-y^TS^T\sqrt{P^{-1}})^T\geq 0$ where $\sqrt{P^{-1}}$ and $\sqrt{P}$ exist due to the fact that $P\geq 0$.