Give two multivariate Gaussian variables $X\sim N(\mu_x,\Sigma_x), Y\sim N(\mu_y,\Sigma_y)$. We have diagonal factorization $\Sigma_x=U_x D_x U_x^{T}=U_xD_x^{1/2}D_x^{1/2}U_x^{T}=U_xD_x^{1/2}(U_xD_x^{1/2})^T=PP^T$, where the columns of $U_x$ are the eigenvectors of $\Sigma_x$ and the diagonal of $D_x=\operatorname{diag}(\lambda_x^1,...,\lambda_x^n)$ contains the eigenvalues of $\Sigma_x$, $D^{1/2}=\operatorname{diag}(\sqrt{\lambda_x^1},...,\sqrt{\lambda_x^n})$. We note the eigenvalues of $\Sigma_y$ as $\lambda_y^1,...,\lambda_y^n$.
I know that there exists a linear transformation $T$ such that $T(X)=Z=P^{-1}(X-\mu_x)\sim N(0,I)$. If we apply the same transformation $T$ on $Y$ and get $Y'=T(Y)$. Then $Y'\sim N(\mu_{y'},\Sigma_{y'})$ is also a Gaussian variable. I also know that $\Sigma_{y'}=P^{-1}\Sigma_y(P^{-1})^T$.
My question is what are the eigenvalues of $P^{-1}\Sigma_y(P^{-1})^T$? What properties can we get about the eigenvalues of $P^{-1}\Sigma_y(P^{-1})^T$ or $\operatorname{tr}(P^{-1}\Sigma_y(P^{-1})^T)$?