TITLE: Proof that the largest eigenvalue of a stochastic matrix is $1$
QUESTION [59 upvotes]: The largest eigenvalue of a stochastic matrix (i.e. a matrix whose entries are positive and whose rows add up to $1$) is $1$.
Wikipedia marks this as a special case of the Perron-Frobenius theorem, but I wonder if there is a simpler (more direct) way to demonstrate this result.

REPLY [3 votes]: If $A \in \textsf{M}_n (\mathbb{C})$, then $\begin{Vmatrix} A \end{Vmatrix}_\infty := \max\limits_{1\le i \le n} \sum_{j=1}^n \vert a_{ij} \vert$ is a matrix norm. Furthermore, $\rho(A) \le \begin{Vmatrix} A \end{Vmatrix}$ for any matrix norm $\vert \vert \cdot \vert\vert$ (e.g., Theorem 5.6.9 in the first-edition of Horn and Johnson's Matrix Analysis). If $A$ is stochastic, then $\begin{Vmatrix} A \end{Vmatrix}_\infty = 1$ so that $\vert \lambda \vert \le \rho(A) \le 1$.
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