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

Let

${}M={\left(a_{ij}\right)}_{1\leq i,j\leq n}\,$

be a real square matrix with non-negative entries.

Then ${}M$ is

column stochastic if and only if ${}M$ is isometric with respect to the sum norm for vectors with non-negative entries, that is,

${}\Vert {Mv}\Vert _{\rm {sum}}=\Vert {v}\Vert _{\rm {sum}}\,$

holds for all

${}v\in \mathbb {R} _{\geq 0}^{n}$.

Proof, Write another proof