Using the Jordan canonical form, show given any square matrix A and e>0, that there exists a matrix Q that
$Q^{-1}AQ=\pmatrix{J^{1}_e&0&...&0\\0&J^{2}_e&...&0\\...&...&...&...\\0&...&...&J^{q}_e}$ where each block $J^{i}_e$ = mI + eN.
I have no idea about how to prove that... I wanted to get some ideas from my previous question but failed. previous one
Hint: if $J$ is an $n$-by-$n$ Jordan block with eigenvalue $\lambda$ and $D = \text{diag}(1,\epsilon,\dots,\epsilon^{n-1})$, compute $D^{-1} J D$.