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Homework Help: Steady state transition matrix

  1. Apr 22, 2010 #1
    Ok, i'm lost. I've an exam coming up so could so with a speedy reply.

    This whole transition matrix stuff is not explained at all in our lecture notes. Here's an example question:

    "Suppose that a country has a fixed number of voters, all of whom vote for
    either party D or party R. Every year, 1/4 of D voters change to party R and 1/3 of R voters switch to party D. Let xn and yn represent the proportions of
    D and R voters respectively after n years (so that xn + yn = 1).
    (a) Find the transition matrix T for this process.
    (b)Explain the term ”steady state”, and find the steady state in this problem.
    (c)Show that xn and yn tend to the steady state values as n goes to infinite, regardless
    of the values of x0 and y0."

    I dont even know what the matrix is suppose to look like...
     
  2. jcsd
  3. Apr 23, 2010 #2
    The matrix is will have: (3/4 1/3) on the top row and (1/4 2/3) on the bottom row. Do you see why? If you multiply this by the vector (D, R) you get the specified voter switches.

    Usually when you are asked to find a steady-state vector one of the eigenvalues of the matrix will be 1, and you need to find the eigenvector corresponding to this eigenvalue. The reason this is called a "steady-state" vector is that the transition matrix does not change it.
     
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