non-negative eigenvector w.r.t. the eigenvalue 1 of a Markov matrix

Question

A non-negative square matrix is called Markov if all the column sums are 1. It is easy to show that any Markov matrix has an eigenvalue 1.

For any Markov matrix, does there always exist an eigenvector associated with the eigenvalue 1 such that all of its entries are non-negative?