I have a matrix of probability coefficients that I have been using to answer some questions about an airline mode. A link to the specifics of the question here. I have been asked to find the equilibrium distribution and prove that it equals the limiting distribution. I have found the limiting distribution by:
$$\lim_{n\to\infty} A^n\pi_0 = (PB^nP^{-1})\pi_0$$
where $\pi_0$ is the initial distribution. This worked out really well and my vector had elements that added up to 1, which gave me peace of mind.
Now I am trying to find the equilibrium distribution.
As I understand it, I should think about the equilibrium distribution as one that does not change from iteration to iteration: $A\pi = \pi$:
So $(A-I)\pi = 0$. No big deal! This is just like solving for the eigenvector for the eiganvalue $\lambda =1$, which I already did in order to find the limiting distribution by way of $P^{-1}AP$!
So the limiting distribution should be equal to the eigenvector corresponding to the eigenvalue of one... Except that it's not. Where is my thinking wrong here?
I simply had to multiply the eigenvector by a scalar in order to get a valid probability distribution, such that the elements of the distribution added to 1.