Consider a discrete-time system where stochastic transitions between the two states (A and B) occur as shown in Figure 2.1. The transition probability in unit time from the state A to B is $\alpha$ and from the state B to A is $\beta$. Note that $0 < \alpha < 1$ and $0 < \beta < 1$. Variables $n$ and $k$ represent discrete time and are integers greater than or equal to 0.

Answer the following questions.

\begin{enumerate}
  \item Let $P_{\mathrm{A}}(n)$ be the probability that the state is A at time $n$ and $P_{\mathrm{B}}(n)$ be the probability that the state is B at time $n$. Let $\boldsymbol{P}(n) = \binom{P_{\mathrm{A}}(n)}{P_{\mathrm{B}}(n)}$. Express matrix $\boldsymbol{M}$ using $\alpha$ and $\beta$, assuming $\boldsymbol{P}(n+1) = \boldsymbol{M}\boldsymbol{P}(n)$.
  \item Obtain all eigenvalues and the corresponding eigenvectors of matrix $\boldsymbol{M}$.
  \item As time tends towards infinity, the probability that the state is A and the probability that the state is B converge towards constant values. Obtain each value.
  \item Assume $R_{\mathrm{A}}(n) = P_{\mathrm{A}}(n) - \lim_{k \rightarrow \infty} P_{\mathrm{A}}(k)$. Express $R_{\mathrm{A}}(n+1)$ by using $R_{\mathrm{A}}(n)$.
\end{enumerate}