We now assume that all coefficients $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ of the stochastic matrix $M$ are strictly positive.
Prove that $\operatorname { dim } \left( \operatorname { ker } \left( M - I _ { n } \right) \right) = 1$.
If $\left( u _ { 1 } , \ldots , u _ { n } \right)$ denotes the components (real) in the canonical basis of a vector of $\operatorname { ker } \left( M - I _ { n } \right)$, one may use $\min _ { 1 \leqslant i \leqslant n } u _ { i }$.