Let $P \in [0,1]^{n \times n}$ be a [irreducible or reducible] stochastic matrix where its rows sum to 1 i.e. $$ \forall i \in \{ 1 , \dots n \} \quad \sum_{j=1}^{n} P_{ij} = 1 $$ It is easy to show that $P$ always has an eigenvalue of 1 by using the above property.
However, I wanted to show that there exists a corresponding left eigenvector [to eigenvalue 1] with real entries, each of which has the same sign [zero entries would be allowed] i.e. non-negative.
Anybody have an idea on how to prove this? Any references would also be greatly appreciated.