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

For the column stochastic ${\displaystyle {}3\times 3}$-matrix

${\displaystyle {\begin{pmatrix}1&0&{\frac {1}{3}}\\0&1&{\frac {1}{3}}\\0&0&{\frac {1}{3}}\end{pmatrix}},}$

the eigenspace to the eigenvalue ${\displaystyle {}1}$ equals ${\displaystyle {}\langle e_{1},\,e_{2}\rangle }$;, 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.