# Complex matrices with null trace

I'm trying to prove the following:

Let $A\in \mathbb{C}^{n\times n}$ be a matrix with null trace; then $A$ is similar to a matrix $B$ such that $B_{jj}=0$ (i.e. it has zeroes on its diagonal).

Any ideas? Induction on $n$ sounded feasible but I wasn't able to put together anything.

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You can read the following short, nice paper

http://www.cs.berkeley.edu/~wkahan/MathH110/trace0.pdf

Please note the gist of the paper for you is Corollary 4: any square matrix over the complex is similar to a matrix all of whose diagonal elements are the same element, and of course this is all you need.

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