Chapter 4.9, Problem 18E

Show that every 2 × 2 stochastic matrix has at least one steady-state vector. Any such matrix can be written in the form $p=\left[\begin{array}{cc}1-\alpha & \beta \\ \alpha & 1-\beta \end{array}\right]$, where α and β are constants between 0 and 1. (There are two linearly independent steady-state vectors if α = β = 0. Otherwise, there is only one.)