I have a transition matrix of $$\begin{bmatrix}0.9&0&0.1\\0.05&0.8&0.2\\0.05&0.2&0.7\end{bmatrix}$$ with a steady state eigenvector of $$c\begin{bmatrix}0.52981294\\0.66226618\\0.52981294\end{bmatrix}$$
These represent the transitions of three populations A, B, and C per month. It is given that there are 10,000 people total in all three populations combined. I need to find the "steady state" number of people in each population that does not change from month to month. How would I go about doing this?
Best Answer
There are three distinct eigenvalues, the largest is one (cf. Perron Frobenius) and the others have modulus less than one.
Suppose $\lambda_1 = 1$, $\lambda_2,\lambda_3$ are the eigenvalues, and $v_k$ the corresponding eigenvectors.
Since it has a full set of eigenvectors the initial population can be written as $p_0 = \sum_k \alpha_k v_k$
The population after $n$ cycles is $p_n=A^n p_0$ where $p_0$ is the initial population of the three groups.
We see that $p_n = \sum_k \alpha_k \lambda_k^n v_k$ and hence $\lim_n p_n = \alpha_1 v_1$.