Let $A$ be an $n \times n$ matrix with real, non-negative entries. Assume $A$ is primitive, meaning there exists an integer $k$ such that $A^k>0$ (here the inequality means all entries in $A$ are positive). Also, assume $A$ is row stochastic, meaning that the the entries of each row sum to 1.
An alternate way of stating the above is to say: let $A$ be the transition probability matrix of an irreducible, aperiodic Markov chain.
Is $A$ diagonalizable? Thank you in advance for any proofs or counterexamples.
For example, consider
$$ \left[ \begin {array}{ccc} 0&{\frac {49}{72}}&{\frac {23}{72}} \\ 1/2&1/6&1/3\\ 1&0&0\end {array} \right] $$
which is not diagonalizable (the eigenvalue $-5/12$ has algebraic multiplicity $2$ but geometric multiplicity $1$).