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.

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- Aug 6th 2013, 11:39 AMHumansimilar to its transpose
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.

- Aug 6th 2013, 01:24 PMDrexel28Re: similar to its transpose
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.