Column stochastic matrix/Positive column/Two-dimensional eigenspace/Example

For the column stochastic $3\times 3$-matrix

$\left(\begin{array}{ccc}1& 0& \frac{1}{3}\\ 0& 1& \frac{1}{3}\\ 0& 0& \frac{1}{3}\end{array}\right),$

the eigenspace to the eigenvalue $1$ equals $⟨e_1,e_2⟩$;, it is two-dimensional. This shows that the conclusion of fact does not hold even when there exists a column (but not a row) with strictly positive entries.