Today I Learned

Some of the things I've learned every day since Oct 10, 2016

47: Invariant Distributions as Eigenvectors

Since a stationary distribution \pi of a finite Markov chain X satisfies \pi P = \pi, where P is the transition matrix of X, it can be seen as an eigenvector of eigenvalue \lambda = 1 under the linear transformation by P. Specifically, \pi is the intersection of the eigenspace E_1 with the hyperplane formed by the constraint that \sum _{i = 1} ^n \pi (i) = 1.

(Here the vector space in question is \mathbb{R}^n, where n is the number of states in X.)

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