Chapter 6.8, Problem 13E

Let A be an $n\times n$ positive stochastic matrix with dominant eigenvalue ${\lambda }_1=1$ and linearly independent eigenvectors $x_1,x_2,...,x_n,$ and let $y_0$ be an initial probability vector for a Markov chain
$y_0,y_1,=A{y}_0,y_2=A{y}_1,...$

1. Show that ${\lambda }_1=1$ has a positive eigenvector $x_1.$
2. Show that ${\Vert y_j\Vert }_1=1,j=0,1,....$
3. Show that if
   $y_0=c_1{x}_1+c_2{x}_2+\cdots +c_n{x}_n$
   then the component $c_1$ in the direction of the positive eigenvector $x_1$ must be nonzero.
4. Show that the state vectors $y_j$ of the Markov chain converge to a steady-state vector.
5. Show that
   $c_1=\frac{1}{{\Vert x_1\Vert }_1}$
   and hence the steady-state vector is independent of the initial probability vector $y_0.$