Question

Let P be the n*n transition matrix of a Markov chain with a finite state space S = {1, 2, ..., n}. Show that 7 is the stationary distribution of the Markov chain, i.e., P = , 2hTi = 1 if and only if (I – P+117) = 17 where I is the n*n identity matrix and 17 = [11...1) is a 1 * n row vector with all components being 1.

Answer 1

First of all note that pi's must be replaced by its transpose, otherwise lacks conformity for multiplication. For further query in above, comment.

the 0 Then Assume that a satities de = a with frial Lesa 2 - I' (1-etha's 'Ia 'e tant aa'-ten 1' [by 0] 7 Į satisbies A6E-P+11) = 1 Next assume that a satistics I'CI-P+?!')=1'', & we shat show that a P = A n osa [Note that all r a normalisation condition, which which is a naturally úi satisfied by any stationary distribution,] Then A (I-P+14) = V => A (I-pj + a 1 = 1 (i alt=1 ='CI-P) +1.1=1 ttle E=A'p as required. to be