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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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    I'm going to perform the unusual move of closing this old question as a duplicate of a recent question. The reason for doing so rather than the other way round is that the other question both is more general (nothing else about the field is needed than having characteristic$~0$), and has a self-contained answer, whereas this one only has a link-only answer.2015-02-09

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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.