TITLE: Are two matrices having the same characteristic and minimal polynomial always similar?
QUESTION [14 upvotes]: If it is not true, can you provide a counter-example?

REPLY [13 votes]: Take the matrices
$$\begin{pmatrix}0&1&0&0\\0&0&0&0\\0&0&0&1\\0&0&0&0\end{pmatrix}\,\,,\,\,\begin{pmatrix}0&1&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}$$
These two matrices have $\,x^4\,$ as char. polynomial and $\,x^2\,$ as minimal one.
Try a nice exercise: prove that the condition is sufficient if the matrix is $\,n\times n\,\,,\,n\leq 3\,$