To solve the problem:

1. Find the eigenvalues of $P$: The matrix $P$ is given as: \frac{1}{3} & \frac{2}{3} \\ \frac{1}{5} & \frac{4}{5} \end{pmatrix}.$$ The eigenvalues are found by solving the characteristic equation: $$\det(P - \lambda I) = 0.$$
2. Find the stationary probability vector $\pi$: A stationary probability vector satisfies $\pi P = \pi$, where $\pi = (\pi_0, \pi_1)$ and $\pi_0 + \pi_1 = 1$.

Let me compute the eigenvalues and the stationary probability vector for you. numpy as np

Define the transition matrix P

P = np.array([[1/3, 2/3], [1/5, 4/5]])

Compute eigenvalues and eigenvectors

eigenvalues, eigenvectors = np.linalg.eig(P)

Normalize the stationary vector (corresponds to eigenvalue = 1)

stationary_vector = eigenvectors[:, np.isclose(eigenvalues, 1)].flatten() stationary_vector = stationary_vector / np.sum(stationary_vector) # Ensure sum is 1

eigenvalues, stationary_vector1. The eigenvalues of the matrix $P$ are: $\lambda_1 = 0.1333, \quad \lambda_2 = 1$

2. The stationary probability vector $\pi$ is: $\pi = \begin{pmatrix} 0.5 \\ 0.5 \end{pmatrix}$

This means that the stationary distribution is uniform over the states.

Would you like further explanations or detailed steps? Here are related questions to explore:

1. What is the physical significance of the eigenvalue $\lambda = 1$?
2. How do the eigenvalues relate to the stability of the Markov chain?
3. What would the stationary distribution look like for a different transition matrix?
4. Can there be more than one stationary probability vector for a given matrix?
5. How does the Markov chain behave as $n \to \infty$?

Tip: Always verify that the sum of the stationary probabilities equals 1 and that they satisfy $\pi P = \pi$.