A question in Dummit & Foote is asking to prove that two $3\times 3$ matrices are similar iff they have the same characteristic and the same minimal polynomial. I was able to prove that. But then the question is asking me to give an explicit counterexample to this assertion for $4\times 4$ matrices. And I do not know how to give this example.
Could anyone help me please?
For example,choose $A=\left( \begin{matrix} 0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ \end{matrix} \right) $, $B= \left( \begin{matrix} 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \end{matrix} \right) $. The characteristic and minimal polynomial is $\lambda^4$ and $\lambda^2$.
But $A$ and $B$ are not similar.