Column stochastic matrix/All to first vertex/Example

The column stochastic ${\displaystyle {}n\times n}$-matrix

${\displaystyle {\begin{pmatrix}1&1&\cdots &1\\0&0&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &0\end{pmatrix}}}$

transforms the distribution ${\displaystyle {}{\begin{pmatrix}v_{1}\\\vdots \\v_{n}\end{pmatrix}}}$ into the distribution

${\displaystyle {}{\begin{pmatrix}1&1&\cdots &1\\0&0&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &0\end{pmatrix}}{\begin{pmatrix}v_{1}\\v_{2}\\\vdots \\v_{n}\end{pmatrix}}={\begin{pmatrix}\sum _{i=1}^{n}v_{i}\\0\\\vdots \\0\end{pmatrix}}={\begin{pmatrix}1\\0\\\vdots \\0\end{pmatrix}}\,.}$

The first standard vector is an eigenvector of the eigenvalue ${\displaystyle {}1}$; every other standard vector, and in fact every distribution vector, is transformed into the first standard vector. The kernel is generated by the vectors ${\displaystyle {}e_{1}-e_{j}}$, ${\displaystyle {}j\geq 2}$, and does not contain any distribution vector.