 Solution

Write $J_k(\lambda )$ for the Jordan block with diagonal $\lambda$ and ones on the superdiagonal. Its characteristic polynomial and minimal polynomial are both $(x-\lambda {)}^k$. Therefore one answer to the first request isFor the second request, takeThe algebraic multiplicities are five and two, and the largest block at each eigenvalue has size two, so both prescribed polynomials hold. They are not similar matrices: the eigenspace at three has dimensions three and four respectively, and similarity preserves eigenspace dimension.

There is no third similarity class. At eigenvalue one, the total size two and maximal block size two force the single block $J_2(1)$. At eigenvalue three, the blocks partition five, each has size at most two, and at least one has size two. The only partitions are $2+2+1$ and $2+1+1+1$. The uniqueness of Jordan normal form up to block order therefore leaves exactly the two classes already exhibited.