Column stochastic matrix/Isometric with respect to sum norm/Fact/Proof

Proof

Let ${}M$ be a column stochastic matrix, and

${}v={\begin{pmatrix}v_{1}\\\vdots \\v_{n}\end{pmatrix}}\,$

be a vector with non-negative entries. Then

${}{\begin{aligned}\Vert {Mv}\Vert _{\rm {sum}}&=\sum _{i=1}^{n}(Mv)_{i}\\&=\sum _{i=1}^{n}{\left(\sum _{j=1}^{n}a_{ij}v_{j}\right)}\\&=\sum _{j=1}^{n}v_{j}{\left(\sum _{i=1}^{n}a_{ij}\right)}\\&=\sum _{j=1}^{n}v_{j}\\&=\Vert {v}\Vert _{\rm {sum}}\end{aligned}}$

holds. If the described isometric property holds, then it holds in particular for the images of the standard vectors. That means that their sum norm equals ${}1$. These images are the corresponding columns of the matrix; therefore, all column sums are ${}1$.

To fact