Solved: Let A be an n n positive stochastic matrix with dominant eigenvalue 1 = 1 and

Chapter 6, Problem 13

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Let A be an n n positive stochastic matrix with dominant eigenvalue 1 = 1 and linearly independent eigenvectors x1, x2, ... , xn, and let y0 be an initial probability vector for a Markov chain y0, y1 = Ay0, y2 = Ay1, ... (a) Show that 1 = 1 has a positive eigenvector x1. (b) Show that yj1 = 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 x11 and hence the steady-state vector is independent of the initial probability vector y0.