Chapter 8, Problem 6P

We will now show that the product of any two $n\times n$ doubly stochastic matrices is doubly stochastic, and that any doubly stochastic $n\times n$ matrix has

$\left[\begin{array}{c}\frac{1}{n}\\ \frac{1}{n}\\ ⋮\\ \frac{1}{n}\end{array}\right]$

as a stable distribution. Let $E_n$ be the $n\times n$ matrix for which each entry is 1.

Show that A is a doubly stochastic $n\times n$ matrix, then

$\left[\begin{array}{c}\frac{1}{n}\\ \frac{1}{n}\\ ⋮\\ \frac{1}{n}\end{array}\right]$

i s a stable distribution for A.Hint: First show that $A\left[\begin{array}{c}1\\ 1\\ ⋮\\ 1\end{array}\right]=\left[\begin{array}{c}1\\ 1\\ ⋮\\ 1\end{array}\right]$