A doubly stochastic matrix

Topic: Discrete-time Markov chains · more questions on this topic
‹ A deterministic three-cycle

Prove that the uniform distribution is stationary for every finite doubly stochastic transition matrix.

Worked answer (click to show)

Let P be m\times m, with every column summing to 1, and let \pi be the row with every entry 1/m. Then

(\pi P)_j=\sum_{i=1}^m \frac{1}{m}p_{ij}=\frac{1}{m}\sum_{i=1}^m p_{ij}=\frac{1}{m}=\pi_j,

because column j sums to 1. So \pi P=\pi.