Let A be an n n positive stochastic matrix with dominant eigenvalue 1 = 1 and linearly independent eigenvectors x1, x2,..., xn, and let

Let A be an n × n positive stochastic matrix with dominant eigenvalue λ1 = 1 and linearly independent eigenvectors x1, x2,..., xn, and let y0 be an initial probability vector for a Markov chain
y0, y1 = Ay0, y2 = A1, ...
(a) Show that λ1 = 1 has a positive eigenvector x1.
(b) Show that ||yj||1 = 1, j = 0, 1,...
(c) Show that if
y0 = c1x1 + c2x2 + ... + cnxn
then the component c1 in the direction of the positive eigenvector x1 must be nonzero.
(d) Show that the state vectors yj of the Markov chain converge to a steady-state vector.
(e) Show that
c1 = 1 / ||x1||1
and hence the steady-state vector is independent of the initial probability vector y0.