Distributive property on trace norm

Question

I hope this is not a trivial question, basically, if we have trace norm of $A$ defined as $||A||_\star := \operatorname{trace}\left(\sqrt{A^*A}\right) = \sum\limits_{i=1}^{\min\{m,n\}} \sigma_i$, if $A=B+C$ and both $B$ and $C$ are p.s.d, do we have $||A||_\star = ||B||_\star + ||C||_\star$?

I think it holds as long as we have distributive property holds on SVD, like $US_AV^T = US_BV^T + US_CV^T = U(S_B+S_C)V^T$.

I think this distributive property also holds, but I am not sure.

EDIT: What if now we restrict $B$ and $C$ to be positive semidefinite?

Answer 1

Hint. The trace norm of a positive semidefinite matrix is merely its trace.