Complex matrices with null trace [duplicate]

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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marked as duplicate by Marc van LeeuwenFeb 9 at 12:32

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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. –  Marc van Leeuwen Feb 9 at 12:32

1 Answer

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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