Problem.
Suppose $A, B$ are both $5 \times 5$ nilpotent complex matrices with the same nullity and the same minimal polynomial. Prove that $A$ and $B$ are similar.
My Question.
Is there a 'clever' way to do this?
I did it by cases: If nullity = 1, 4, or 5, then only 1 JCF is possible. If nullity = 2, then there are two possible JCFs, but they have different minimal polynomials; same if nullity = 3. So in short, if $A$ and $B$ share the same nullity and minimal polynomial, then they share the same JCF. So, does this seem like a correct approach? And also, anytime I'm doing something by cases, I always wonder if there's a more elegant way of handling all the cases at once?
Thanks a ton!