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

{{ Mathematical text/Proof |Text= {{ Proofstructure |Strategy= |Notation= |Proof= Let ${\displaystyle {}w\in \mathbb {R} ^{n}}$ be the stationary distribution, which is uniquely determined because of fact  (3). We set

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

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

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

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

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

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

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

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

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

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

we can write

${\displaystyle {}v=w+u\,}$

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

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

and the reasoning before, this sequence converges to ${\displaystyle {}w}$. |Closure= }} |Textform=Proof |Category=See }}