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Find a finite set of $n\times n$ matrices (specific number) with complex elements so that every nilpotent $n\times n$ matrix with complex elements is similar to only one of them.

I tried to think a general form of nilpotent matrix but I can't find one. Also I think that every nilpotent is similar to other nilpotents so how can it be finite set? Any hints?

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Hint: What can you say about the Jordan normal form of a nilpotent matrix?

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  • $\begingroup$ Eigenvalues are zero so jordan is also a nilpotent. You mean the set I look for is the jordan matrices? $\endgroup$ – user285936 Feb 14 '18 at 12:17
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    $\begingroup$ @GeorgeF Yes. Think about it. For instance, when $n=2$, one matrix is enough. And when $n=3$, $2$ matrices are enough. $\endgroup$ – José Carlos Santos Feb 14 '18 at 12:23
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    $\begingroup$ @JoséCarlosSantos Won't $n=2$ need two matrices? Since the Jordan normal form of a nilpotent matrix can be either $\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ or $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, and these two aren't similar. $\endgroup$ – lisyarus Feb 14 '18 at 12:31
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    $\begingroup$ @lisyarus Sure! I forgot the null matrix. And when $n=3$, $3$ matrices are enough. $\endgroup$ – José Carlos Santos Feb 14 '18 at 12:33

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