Let $X,Y\in R^{n\times p}$, $\Lambda = \operatorname{diag}(\lambda_1,\lambda_2,\cdots,\lambda_n)$, $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n\gt 0$, $\Phi=\operatorname{diag}(\mu_1,\mu_2,\cdots,\mu_p)$, $\mu_1\geq\mu_2\cdots\geq\mu_p\gt0$. Show the following inequality holds
$\operatorname{tr}(X^T X Y^{T}\Lambda Y\Phi) \geq \lambda_n\operatorname{tr}(X^T X Y^{T} Y\Phi)$
Can anybody help me out with this proof? Thank you in advance