Let $A\in[0,1]^{n\times n}$ be a row-stochastic matrix. Then how can I prove that the induced norm of $A$ with respect to $\lVert\cdot\rVert_{\infty}$ is equal to one?
In other words, I have to prove that, \begin{equation} \lVert A \rVert_{\infty \rightarrow \infty} = sup \frac{\lVert Ax\rVert_{\infty}}{\lVert x\rVert_{\infty}}=1 \end{equation} where $x\in\mathbb{R}^n$\ $\{0\}$.
Each coordinate of $Ay$ is a convex combination of the coordinates of $y$. So, $\|Ay\|_\infty \leq \|y\|_\infty$ for each $y$.
Then, choosing $y$ with all coordinates equal to some constant $\alpha$, we see that $\|Ay\|_\infty = \|y\|_\infty$ since $A$ is row stochastic. This gives us a lower bound on the supremum.
The first part gives us that $\sup_x \frac{\|Ax\|_\infty}{\|x\|_\infty} \leq 1$ while the second part shows that $\sup_x \frac{\|Ax\|_\infty}{\|x\|_\infty} \geq 1$. Putting these two inequalities together gives equality.