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Let $A$ be an $n\times n$ complex matrix with $trace(A)=0$. We need to Show that $A$ is similar to a matrix with all $0'$ s along the main diagonal.

What I thought: $A$ is not zero matrix, also $A\ne cI$ for any $c\in\mathbb{C}^*$

so $\exists v\in \mathbb{C}^n$ which is not an eigen vector of $A$, then I thought of taking a basis whose first two vectors are $v,Av$ and then with respect to this basis what will be the matrix $A$?

Thanks for helping.

Myshkin
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  • Mathematical induction! Can you turn the last (i.e. bottom right) entry of $A$ into zero via similarity transform? – user1551 Jan 11 '15 at 09:15
  • You can make $A$ traceless by subtracting off a zero matrix $D$ with bottom-right entry equal to $\mbox{tr}(A)$. Then it suffices to prove $A-D$ is similar to a matrix of all zeros. You can find a proof here: https://math.stackexchange.com/a/252324/22064 – Alex R. Mar 18 '19 at 18:22

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