Two matrices with same characteristic polynomial need not be similar.

We all know that any two similar matrices have the same characteristic polynomial. Naturally, we might ask for the converse. Unfortunately, the converse is not true and a counterexample on can find at the following link,

http://www.mathcounterexamples.net/two-non-similar-matrices-having-same-minimal-and-characteristic-polynomials/

Top of the place for the answer, above site, in my opinion, seems to be a great site to find counterexamples in various disciplines.

Well, as I thought, some of you might be thinking that $A$ and $A^T$ might serve as a counterexample. In fact, $A$ and $A^T$ are similar always. One can see this on,

https://math.stackexchange.com/questions/62497/matrix-is-conjugate-to-its-own-transpose

As explained on the link, if $A$ happens to be diagonalizable it is extremely easy to conclude.