# Matrices similar to nilpotent

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?

• @GeorgeF Yes. Think about it. For instance, when $n=2$, one matrix is enough. And when $n=3$, $2$ matrices are enough. – José Carlos Santos Feb 14 '18 at 12:23
• @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. – lisyarus Feb 14 '18 at 12:31
• @lisyarus Sure! I forgot the null matrix. And when $n=3$, $3$ matrices are enough. – José Carlos Santos Feb 14 '18 at 12:33