If $v$ is a left eigenvector of stochastic matrix $P$ with $vP = \lambda v$ for $\lambda <1$, can you show that $\sum_{i = 1}^{N} v_{i} = 0$. You can assume that $v$ is normalized.
2026-03-26 07:57:41.1774511861
Left eigenvector for eigenvalue < 1 for a square stochastic matrix: coordinates of eigenvector sum to zero.
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Hint. Consider $vPe$ where $e=(1,1,\ldots,1)^T$.