Is the following inequality true?
$$\mbox{Tr} \left( \mathrm P \, \mathrm M^T \mathrm M \right) \leq \lambda_{\max}(\mathrm P) \, \|\mathrm M\|_F^2$$
where $\mathrm P$ is a positive definite matrix with appropriate dimension. How about the following?
$$\mbox{Tr}(\mathrm A \mathrm B)\leq \|\mathrm A\|_F \|\mathrm B\|_F$$
Best Answer
$$\mbox{tr}(\mathrm A \mathrm B) = \left\langle \mathrm A^\top, \mathrm B \right\rangle = \langle \mbox{vec} (\mathrm A^\top), \mbox{vec} (\mathrm B) \rangle \leq \|\mbox{vec} (\mathrm A^\top)\|_2 \|\mbox{vec} (\mathrm B)\|_2 = \|\mathrm A\|_F \|\mathrm B\|_F$$