If the trace of the product of two matrices is zero,is the product of the matrices zero?

Question

Does $\:\operatorname{ Tr}(XY) = 0 \Rightarrow XY=O $, for all $X,Y \succeq O$ ?

Answer 1

Hint: Let $A$ and $B$ be the positive semi-definite square roots of $X$ and $Y$ respectively, i.e. $X = A^2$ and $Y = B^2$ where $A,B \succeq O$. Try to prove $\text{tr}(XY) = \|AB\|_F^2$ (the cyclic property of the trace might help). After doing this, if $\text{tr}(XY) = 0$, what can you say about $AB$, and consequently $XY$?

Answer 2

Hint: If $X = \sum_{i}x_ix_i^*$ and $Y = \sum_{j}y_jy_j^*$, then $$ XY = \sum_{i,j} (x_i^*y_j)x_iy_j^*, \quad \operatorname{tr}(XY) = \sum_{i,j} |x_i^*y_j|^2. $$