Let N be an k*k nilpotent matrix with nilpotency index k i.e. $\displaystyle N^k=0$ but $\displaystyle N^{k-1} \neq 0$ . I want to show that $\displaystyle N^t$ (transpose of N) is similar to N.
I mean, since $\displaystyle \mathbb{R}$ or $\displaystyle \mathbb{C}$ contain all the eigenvalues of $\displaystyle N$ you can conjugate to put it in Jordan canonical form and then you only have to note that the Jordan matrix is similar to its transpose.