# 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 $X$.)