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Homework Help: Stochastic matrix

  1. Oct 12, 2009 #1
    1. The problem statement, all variables and given/known data
    Let P be a stochastic matrix on a finite set I. Show that a distribution π is invariant for P
    if and only if π(I-P+A) = a, where A = (aij : i,j in I) with aij = 1 for all i and j, and a = (ai : i in I) with ai = 1 for all i. Deduce that if P is irreducible then I-P+A is invertible.
    Note that this enables one to compute the invariant distribution
    by any standard method of inverting a matrix.

    2. Relevant equations

    3. The attempt at a solution
    I have no idea how I approach to the answer.
    Anybody give me some idea.
  2. jcsd
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