Positive matrices are diagonalizable in $\mathbb C$?

Question

Let $A$ be a square matrix with real positive entries. Is it true that it is diagonalizable in $\mathbb C$?

My guess is that it is false, but the counterexample can't be a $2\times 2$ matrix, since I checked that the conditions on trace and determinant lead to an absurd.

Is there a well-known counterexample? Like something that comes from Markov Chains or Dynamical Systems?

(it comes from this question on perturbations of matrices)

Answer 1

After computation the counter-example is $\begin{pmatrix}1&2&3\\1&2&3\\6&2&8\end{pmatrix}$

Answer 2

Here is an example, derived from the "telephone matrix": $$\begin{pmatrix}1&2&3\\4&5&6\\1&2&3\end{pmatrix}$$ Its characteristic polynomial is given by $t^2(t-9)$.