Given constants $\alpha\geq \beta>0$, find a nonsingular matrix $\Psi\in\mathbb{R}^{n\times n}$ and a postive definite matrix $P$ such that $$ \begin{aligned} \alpha P\geqslant \Psi^TP\Psi&\geqslant \beta P\\ \end{aligned} $$ It is preferred that the method devised to find $\Psi,P$ is based on LMI, cone complentarity linearization or BMI or SDP programming. Here $\geq$ means "positive semidefinite" for matrices. Any help is appreciated!
And, there is an extended question about this problem, what kind of property should $\Psi$ be with such that $\alpha P\geq \Psi^TP\Psi\geq \beta P$ holds for all positive semidefinite $P$?