Let $A, B$ be two $n \times n$ matrices over a field $F$ and $A,B$ have the same characteristic and minimal polynomials. If no eigenvalue has algebraic multiplicity greater than $6$ and the solution space of $A-c_iI$ and $B-c_iI$ are same then $A$ and $B$ are similar (characteristic polynomial of $A,B$ is $f = (x - c_1)^{d_1} \cdots (x - c_k)^{d_k}$).
I have to use the following result.
If $A, B$ are two $6 \times 6$ nilpotent matrices, then $A, B$ are similar if and only if they have same minimal polynomial and the same nullity.
These are exercises from Hoffman and Kunze. I have proved the second result. I am not sure how to use this to prove the first result. I was thinking of breaking it into cases but that is becoming too long. Any hint would be appreciated rather than a complete answer.