The transition matrix in Example 5 has the property that both its rows and its columns all add up to I. In general. a matrix A is said to be doubly stochastic if both A and AT are stochastic. Let A be an ii x ii doubly stochastic matrix whose eigenvalues satisfy
λ1 = 1 and |λ1| < 1 for j = 2,3,......n
If e is the vector in Rn whose entries are all equal to 1, show that the Markov chain will converge to the steady slate vector x = for any starting vector X. Thus, I/n e or a doubly stochastic transition matrix, the steady-state vector will assign equal probabilities to all possible outcomes.