1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook