Would the results of parts (c) and (d) in Exercise 13 be valid if the stochastic matrix A was not a positive matrix? Answer this same question in the case when A is a nonnegative stochastic matrix and, for some positive integer k, the matrix Ak is positive. Explain your answers.
Exercise 13
(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.