Some equivalent transform about trace?

If rank(X)>1 or X is full rank, can we have similar equivalent transmfrom like below two equations?

thank you, in my problem, X is a PSD matrix, which needs to be optimized. I think i can not perform matrix decomposition of it.

ok, thank you.
i want to tackle the robust constraint 42b,where w is the optimization problem.

ok,thank you, i think i should read some linear algebra relared books to tackle my problem.