Given two 6×6 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 7×7.
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 7×7 case is any different.
Best Answer
Note that two matrices are similar if and only if they have the same Jordan canonical form. With that in mind, note that the Jordan forms of the matrices we're discussing:
In the $6 \times 6$ case, this is enough to ensure that the Jordan forms are the same (up to a permutation of the blocks). In the $7 \times 7$ case, we have counterexamples such as the pair $$ \pmatrix{ 0&1\\&0&1\\&&0\\ &&&0&1\\ &&&&0&1\\ &&&&&0\\ &&&&&&0}, \quad \pmatrix{ 0&1\\&0&1\\&&0\\ &&&0&1\\ &&&&0&\\ &&&&&0&1\\ &&&&&&0} $$