Why does $\mathrm{Rank}(A{A^*} - {A^*}A) \ne 1$? Given $A \in M_n$, why does $\mathrm{Rank}(A{A^*} - {A^*}A) \neq 1$?

More generally, let $M\in M_n(\mathbb{C})$ s.t. $rank(M)=1,tr(M)=0$. Then $M$ is similar to $diag(\begin{pmatrix}0&1\\0&0\end{pmatrix},0_{n-2})$. 
Proof: According to Jordan theory, $rank(M)=1$ implies that $M$ is similar to $diag(\begin{pmatrix}0&1\\0&0\end{pmatrix},0_{n-2})$ or to $diag(a,0_{n-1})$ where $a\not= 0$. Since $tr(M)=0$, the second case is impossible.
