I would like to show that every trace zero square matrix is similar to one with zero diagonal elements.
This question has been asked before, and has had an answer by Don Antonio.
And my problem is that I cannot understand the cited paper.
In the cited paper, (proof 4), one finds the sentence
Since $\text{Tr}(K) = \text{Tr}(B^{–1}SB) = \text{Tr}(S) = 0$, this step can be repeated to replace $K$ by a matrix whose every diagonal element is zero ( thereby changing $c$ and $r^T$ ) thus constructing $C$ so that every diagonal element of $C^{–1}SC$ is zero.
Question
I thought that, since $\text{Tr}(K)=0$, we can, by the induction hypothesis, find an invertible $D$, such that $D^{-1}KD$ has diagonal elements $=0$. But how does this enable us to replace $K$ by a matrix with zero diagonal elements, at the cost of changing $r^T$ and $c$?
I tried to constuct some matrix $C$ from $D$ such that $C^{-1}\begin{bmatrix}0&r^T\\c&K\end{bmatrix}C=\begin{bmatrix}0&r'^T\\c'&K'\end{bmatrix}$ with $K'$ having zero diagonal elements, but to no avail.
I have tried various choices of $C$, but the result of the multiplication refuses to be of the required form, so I wonder if I am missing something here?
Any hint is well-appreciated.
