An Illustrative Example: Limiting Probabilities

• If we use each of the instructions with equal probability the Markov transition matrix is the average of all 8, i.e.
  
  $$M = 1/8 \left( \begin{array}{cccc} 4 & 2 & 2 & 0 \\ 2 & 4 & 0 & 2 \\ 2 & 0 & 4 & 2 \\ 0 & 2 & 2 & 4 \\ \end{array}\right)$$
  
  

• $u_\infty = 1/4 (1,1,1,1)$

  The eigenvalues $\lambda$ and corresponding eigenvectors E of M are
  

  $$\begin{array}{r@{ = }l@{% (}rrrr@{\hspace{\tabcolsep} )}} \la... ...& 1 & 1 & 1\\ \lambda_{11} & 0 & 1 & -1 & -1& 1\\ \end{array}$$
  
  

  Note since M is symmetric the other eigenvalues are also real.