Chapter 6.3, Problem 28E

The transition matrix in Example 5 has the property that both its rows and its columns add up to 1. In general, a matrix A is said to be doubly stochastic if both A and $A^T$ are stochastic. Let A be an $n\times n$ doubly stochastic matrix whose eigenvalues satisfy
${\lambda }_1=1$ and $\left|{\lambda }_j\right|<1$ for $j=2,3,...,n$
Show that if e is the vector in ${ℝ}^n$ whose entries are all equal to 1, then the Markov chain will converge to the steady-state vector $x=\frac{1}{n}e$ for any starting vector $x_0.$ Thus, for a doubly stochastic transition matrix, the steady-state vector will assign equal probabilities to all possible outcomes.