I'm reading through my textbook, Introduction to Stochastic Processes (Lawler), before the semester begins in hopes of getting ahead, and I've run into something I just plain cannot figure out: How to compute the invariant probability vector for a transition matrix. I was hoping that one (or many) of you would be able to walk me through how you would do this for just a simple matrix:
$\begin{bmatrix} .4&.2&.4 \\\\ .6&0&.4 \\\\ .2&.5&.3 \end{bmatrix}$
I know that you can compute it by raising the matrix to a large power, but this practice problem says to "compute the invariant probability vector as a left eigenvector." How would one go about doing this?
Thanks for your help!