Prove that $J_{n}(0)$ and $(J_{n}(0))^t$ are similar ($J_n(0)$ is a $n \times n$ Jordanian block which belongs to the eigenvalue $0$).
Use your answer and Jordanian form to prove that every matrix $A \in M_{nxn}^C$ is similar to $A^t$.
My problem is with the first question. Let's take for example the following matrices from a Jordanian form:
$$ B = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{bmatrix} $$
$$ B^2 = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} $$
Who's ranks are 1 and 2 respectively. Similar matrices should have the same rank, therefore $B$ and $B^2$ aren't similar and the first question can't be proven right. Since I didn't prove the first question, I guess I can't continue to the second one.
I probably missed something, any help would be appriciated, thanks!
The minimal polynomial of $J_n(0)^t$ is $x^n$. This is its characteristic polynomial as well. This tells you that the Jordan canonical of $J_n(0)^t$ is $J_n(0)$. But then they are similar.