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