Given two 6x6 nilpotent matrices with the same minimal polynomial and same rank, prove they're similar. Also prove that this is not necessarily the case if the two matrices are 7x7.
If two matrix have the minimal polynomial and same rank, then the following can be generalized:
1) they have the same eigenvalue, 0
2) then have the same nilpotent index
3) they have the same geometric multiplicity
But I'm not seeing how this can explicitly imply similarity and how the 7x7 case is any different.