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Linear Algebra; Stochastic matrix and Steady State vectors

  1. Nov 18, 2008 #1
    1. The problem statement, all variables and given/known data

    Question: 18. Show that every 2 x 2 stochastic matrix has at least one steady-state vector. Any such matrix can be written as

    P = |1-a b |
    | a 1-b |

    where a and b are constants between 0 and 1. (There are two linearly independent steady-state vectors if a = b = 0. Otherwise, there is only one.)


    3. The attempt at a solution

    I am guessing that they want me to show that the above matrix has a solution to the equation Px = x so that the steady state vector exists since there is a solution. My solution was that since a and b are other constants between 0 and 1 the matrix would have a solution?

    Thank you for the help!
     
  2. jcsd
  3. Nov 19, 2008 #2
    Hi Mustang,

    Consider that if Px = x, then Px - x = 0, i.e., (P - I)x = 0, where I is the 2x2 identity matrix. As a steady state vector (just a certain type of "eigenvector", if you are familiar with the term) is necessarily nonzero, recall how one can use the determinant to determine where the system Ax=0 has a nonzero solution.
     
    Last edited: Nov 19, 2008
  4. Nov 19, 2008 #3
    Ah!!! I see what you mean, thanks a lot Unco, much appreciated!
     
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