Editing Markov chain (section)

====Stationary distribution relation to eigenvectors and simplices====
A stationary distribution {{pi}} is a (row) vector, whose entries are non-negative and sum to 1, is unchanged by the operation of transition matrix '''P''' on it and so is defined by
:<math> \pi\mathbf{P} = \pi.</math>

By comparing this definition with that of an [[eigenvector]] we see that the two concepts are related and that
:<math>\pi=\frac{e}{\sum_i{e_i}}</math>
is a normalized (<math display="inline">\sum_i \pi_i=1</math>) multiple of a left eigenvector '''e''' of the transition matrix '''P''' with an [[eigenvalue]] of 1. If there is more than one unit eigenvector then a weighted sum of the corresponding stationary states is also a stationary state. But for a Markov chain one is usually more interested in a stationary state that is the limit of the sequence of distributions for some initial distribution.

The values of a stationary distribution <math> \textstyle \pi_i </math> are associated with the state space of '''P''' and its eigenvectors have their relative proportions preserved. Since the components of &pi; are positive and the constraint that their sum is unity can be rewritten as <math display="inline">\sum_i 1 \cdot \pi_i=1</math> we see that the [[dot product]] of &pi; with a vector whose components are all 1 is unity and that &pi; lies on a [[standard simplex|simplex]].