Some one can help me with this problem?
I have two real positive-semidefinite matrices $P$ and $Z$, $P \succeq 0$, $Z \succeq 0$, and they are both symmetric ($P^T = P$ and $Z^T = Z$). Also trace$ (P\cdot Z) = 0$. Does that mean $P \cdot Z = 0$ ? Assume they are both $n \times n$ square matrices. Thanks！
Best Answer
Since $P$ and $Z$ are semidefinite, $P=S^TS$ and $Z=YY^T$ for some $S$ and $Y$. Then $$ \mathrm{tr}(PZ)=\mathrm{tr}(S^TSYY^T)=\mathrm{tr}(Y^TS^TSY)=\mathrm{tr}[(SY)^T(SY)]=\|SY\|_F^2. $$ Hence $SY=0$ and $PZ=S^T(SY)Y^T=S^T(0)Y^T=0$.