Find Jordan form of matrix with parameter $m\in\mathbb R$

Question

the (real) matrix is $\left[ \begin{matrix} 0& 0& 0\\ 1& 0& -m^{2}\\0& 1& m\\\end{matrix} \right]$. Well, the characteristic polynomial is $p(t)=t(t^2-mt+m^2)$ so it doesn't have any real roots except 0, and then I got stuck.

Answer 1

For the case in which $m \neq 0$, the matrix has three distinct eigenvalues. So, the Jordan form is just the diagonal matrix with those eigenvalues on the diagonal.

For the case in which $m = 0$, we note that transposing the matrix produces a Jordan-form matrix. Any matrix is similar to its transpose.