Why is the eigenvalue of a stochastic matrix always $1$? I have found lots of articles simply saying it is obvious that the eigenvalue is $1$ but can't get my head around the proofs.
2026-03-26 01:01:49.1774486909
Eigenvalue of stochastic matrix
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If the rows of $\mathrm P \in \mathbb R^{n \times n}$ sum to $1$, then
$$\mathrm P 1_n = 1_n$$
Hence, $(1,1_n)$ is an eigenpair of $\mathrm P$.