Column stochastic matrix/Positive row/Convergence/Fact/Proof

Proof

Let ${}w\in \mathbb {R} ^{n}$ be the stationary distribution, which is uniquely determined because of fact  (3). We set

${}U={\left\{{\begin{pmatrix}u_{1}\\\vdots \\u_{n}\end{pmatrix}}\mid \sum _{i=1}^{n}u_{i}=0\right\}}\subseteq \mathbb {R} ^{n}\,.$

This is a linear subspace of ${}\mathbb {R} ^{n}$ of dimension ${}n-1$. Due to fact  (2), ${}w$ has only non-negative entries; therefore, it does not belong to ${}U$. Because of

${}{\begin{aligned}\sum _{i=1}^{n}(Mu)_{i}&=\sum _{i=1}^{n}{\left(\sum _{j=1}^{n}a_{ij}u_{j}\right)}\\&=\sum _{j=1}^{n}u_{j}{\left(\sum _{i=1}^{n}a_{ij}\right)}\\&=\sum _{j=1}^{n}u_{j},\end{aligned}}$

${}U$ is invariant under the matrix ${}M$. Hence,

${}V=\mathbb {R} w\oplus U\,$

is a direct sum decomposition into invariant linear subspaces. For every ${}u\in U$ with ${}\mid \mid \!u\!\mid \mid _{\operatorname {sum} }=1$, we have

${}\mid \mid \!Mu\!\mid \mid _{\operatorname {sum} }<1\,$

due to fact  (2). The sphere of radius ${}1$ is compact with respect to every norm; therefore, the induced maximum norm of ${}M{|}_{U}$ is smaller than ${}1$. Because of fact and fact, the sequence ${}M^{n}u$ converges for every ${}u\in U$ to the zero vector.

Let now ${}v\in V$ be a distribution vector; because of

${}\sum _{i=1}^{n}v_{i}=1=\sum _{i=1}^{n}w_{i}\,,$

we can write

${}v=w+u\,$

with ${}u\in U$. Because of

${}M^{m}v=M^{m}(w+u)=M^{m}w+M^{m}u=w+M^{m}u\,,$

and the reasoning before, this sequence converges to ${}w$.

To fact