Column stochastic matrix/Positive row/One-dimensional eigenspace/Fact

Let ${}M$ be a column stochastic matrix. Then the following statements hold.

1. There exists eigenvectors to the eigenvalue ${}1$.
2. If there exists a row such that all its entries are positive, then for every vector ${}v\in V$ that has a positive and also a negative entry, the estimate

   ${}\mid \mid \!Mv\!\mid \mid _{\operatorname {sum} }<\mid \mid \!v\!\mid \mid _{\operatorname {sum} }\,$

   holds.

3. If there exists a row such that all its entries are positive, then the eigenspace of the eigenvalue ${}1$ is one-dimensional. There exists an eigenvector where all entries are no-negative; in particular, there is a uniquely determined stationary distribution.

Proof, Write another proof