I have the two problems below from a practice exam. I can prove them on their own but am not exactly sure if/how to show that they only hold for symmetric matrices and for '3)' showing that it only holds for a matrix with only positive eigenvalues. I know that if the eigenvalues are all positive the determinant will be positive and the trace but cant see how that affects whether '3)' is true or not.
- Show that $\operatorname{Tr}(A^2) \leq \operatorname{Tr}(A)^2$ holds for any symmetric matrix $A$ whose eigenvalues are all non-negative.
- Show that $\operatorname{Tr}(AB)^2 \le \operatorname{Tr}(A^2)\operatorname{Tr}(B^2)$ holds for any symmetric matrices $A$ and $B$.