Let we have the estimate $$ {\rm tr}(P)\ge\gamma>0, $$ where $P\in\mathbb R^{n\times n}$ is positive-definite matrix.
Is it possible to obtain the nontrivial lower bound in terms of $P$ and $C$ for $$ {\rm tr}(CPC^{\rm T}), $$ where $C\in\mathbb R^{m\times n}$ and $m<n$?
The estimate $$ {\rm tr}(CPC^{\rm T})={\rm tr}(PC^{\rm T}C)\ge\lambda_{\max}(P)\cdot\lambda_{\min}(C^{\rm T}C) $$ gives us the trivial zero bound only.