Similarity of two matrices with the same characteristic polynomial and minimial polynomial

Question

Suppose $A,B\in \mathbb{M_n}$ where $n\in \mathbb{Z_{n>0}}$. If A and B both have i) same characteristic polynomial, ii) same minimal polynomial and iii) multiplicity of each eigenvalue in the characteristic polynomial not greater than 3, prove that they are similar.

Answer 1

HINT:

Considering the sizes of Jordan cells associated to a given eigenvalue, you are reduced to showing:

Let $1\le a_1 \le \ldots \le a_k$, $1\le b_1 \ldots \le b_l$ be integers such that $$\sum a_i = \sum b_j\le 3\\ \max a_i = \max b_j $$ Then $k=l$ and $a_i = b_i$ for all $i$. There are but three cases to consider, according to what that $\max \in \{1,2,3\}$ is.