[Math] Similar matrices with the same minimal polynomial

Question

For which $n$ is it true that two $n \times n$ matrices are similar if they have the same minimal polynomial and the same characteristic polynomial?

Answer 1

Counterexample for $n = 4$: $$\begin{pmatrix} 2 & 1 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2 \end{pmatrix}, \begin{pmatrix} 2 & 1 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{pmatrix}.$$ From reading down the diagonal, both of these have only $\lambda = 2$ as an eigenvalue with multiplicity $4$. Computation reveals that $(z - 2)^2$ is the minimal polynomial (can be seen by the fact that $2$ is the size of the largest Jordan Block).

But, they are not similar, as the former has an eigenspace of dimension $2$, whereas the latter has an eigenspace of dimension $3$.