Complex matrices with trace=0

Question

Let $A$ be a $n\times n$ matrix with complex entries such that $\operatorname{Trace}(A) = 0$. Show that $A$ is similar to a matrix with $0$'s in the diagonal entries. I think I have to use Schur's Lemma , but this is not helping me much .

Thanks for any help .

Answer 1

I could write the answer but I think it's better for you to check directly here http://www.cs.berkeley.edu/~wkahan/MathH110/trace0.pdf . There are there some references to nice generalizations.