From Hoffman and Kunze.
Let $N_1$ and $N_2$ be 6 X 6 nilpotent matrices over the field F. Suppose that $N_1$ and $N_2$ have the same minimal polynomial and the same nullity. Prove that $N_1$ and $N_2$ are similar. Show that this is not true for 7 X 7 nilpotent matrices.
Right so nilpotent matrices have $N^k= 0$ for some k. Since these have the same nullity and minimal polynomial $N_1^k = N_2^k = 0$ from some k right? The same minimal polynomial implies the same characteristic values but thats not enough to say they have the same jordan form and are thus similar right?
and how would I get started in proving this breaks for 7x7 nilpotent matrices?
Thanks in advance for any assistance.