By a stationary distribution for a random process with transition matrix P we mean a system state vector π which remains unchanged after operating on it with P, i.e., Pπ=π. Given P, how do can we determine π? A simple method is to compare the definition of the stationary state with the definition of a eigenvector and deduce a relation connecting the state vector with the eigenvector.
Setting the eigenvalue μ=1 makes the two equations nearly identical. So if we can find an eigenvector of P which has an eigenvalue equal to 1 then we can find the direction of π. To find the magnitude in the given direction we use the fact that the sum of the components of π is one to determine the multiplier for e. We can illustrate how this works by determining the stationary state for a simple random walk problem. In the state diagram below q=1-p:
Looking at the eigenvectors for the transition matrix we see that there are two with eigenvalues equal to 1.
Here each column of π is a stationary state corresponding to final states 1 and 2.