Steady State

The key to finding the steady state is that we are solving for the vector $\vec q$ for stochastic matrix $P$ where

\[P_{\vec q} = \vec q\]

Which is the same as

\[(P-I)\vec q = 0\]

To find the steady state vector given a stochastic matrix:

1. Set the eigenvalue $\lambda$ to one.
2. Find the matrix $A-\lambda I$.
3. You may scale it as you please so long as you normalize afterwards.
4. Put it into the form $A-\lambda I = \vec 0$.
5. Find the free variables and put it in parameter form.
6. That should get you the steady state vector.