Steady state transition matrix

[Elpmek]

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 don't even know what the matrix is suppose to look like...

 

[mrbohn1]

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.