Show that a stochastic matrix always has a row eigenvector with eigenvalue 1. Sh

Question

Show that a stochastic matrix always has a row eigenvector with eigenvalue 1. Show that this row eigenvector can be chosen to correspond a stationary distribution of the induced Markov chain. Hint: It is an easy consequence of the Brouwer Fixed Point Theorem,which says, inter alia, that any continuous map from a compact convex set to itself must have a fixed point. Show that a stochastic matrix always has a row eigenvector with eigenvalue 1. Show that this row eigenvector can be chosen to correspond a stationary distribution of the induced Markov chain. Hint: It is an easy consequence of the Brouwer Fixed Point Theorem,which says, inter alia, that any continuous map from a compact convex set to itself must have a fixed point.

Explanation / Answer

Let given stochastic matrix is denoted by A where

A=[aij] - matrix over R

0aij1 ij

jaij=1 i

i.e the sum along each column of A is 1

Since A describes a transition from some vector that encodes probability distribution to another vector that also encodes probability distribution, A is not allowed to scale up or scale down the vector along the same direction, because otherwise that vector will no longer have all its entries add up to 1 and it would no longer be a probability distribution.

In other words, suppose A is a markov matrix, and v is a vector that represents some probability distribution. This means that all the entries in v have to add up to one.

Now let v=vA. If v were an eigenvector, that means that v=v. Since v is also a probability distribution, all its entries must also add up to 1, and this can only happen if =1

Answer

TY!

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