When is a positive stochastic matrix diagonalizable?

Question

I have a positive matrix $W\in\mathbb{R}^{n\times n}$ and construct a stochastic matrix, $A=D^{-1}W$ where $D$ is diagonal and $D_{ii}=\sum_{j=1}^n W_{ij}$. I am trying to figure out when this matrix $A$ is diagonalizable. From Markov chain theory, I know that if $A^T$ satisfies detail balance (is reversible) then it is diagonalizable, but without knowing about Markov chains, is there any condition we can come up with for $A$ to be diagonalizable? Thanks in advance.