I have the following transition matrix of a markov chain: $$\mathbb{P}= \begin{bmatrix} 1 & 0 & 0 & 0 & 0\\ q & \delta & p & 0 & 0\\ 0 & q & \delta & p & 0\\ 0 & 0 & 0 & \delta+q & p\\ 0 & 0 & 0 & q & \delta+p \end{bmatrix}$$ I am trying to calculate the limit of $\mathbb{P}^{n}$.
There are two closed sets $\{1\}$ and $\{4,5\}$. In addition, these two sets are irreducible. States $2$ and $3$ communicate those two closed sets, they will get lost in those two sets eventually. So, $p_{2,2}=p_{2,3}=p_{3,2}=p_{3,3}=0$ in the long run. State $1$ is a closed state, it stays there forever. For the rest of the entries, I applied the kolmogorov equation.
There are a lot of computations when using the kolmogorov equation. I am wondering if there are better ways.
With a heavy calculation (that's the crux of this problem), you can verify that $\mathbb{P}$ has the following Jordan canonical form (a more secured and expedient way is to use Wolfram Alpha Jordan Form Calculator): \begin{equation} J = \text{diag}(1, \delta, \delta - \sqrt{pq}, \delta + \sqrt{pq}, 1), \end{equation} i.e., there exists an invertible matrix $Q$, such that $\mathbb{P} = QJQ^{-1}$. One choice of $Q$ is as follows: \begin{equation} Q = \begin{bmatrix} q^{-2}\alpha & 0 & 0 & 0 & 0 \\ q^{-1}(1 - \delta) & \rho^2 & -\sqrt{\rho} & \sqrt{\rho} & \alpha^{-1}p^2 \\ 1 & 0 & 1 & 1 & \alpha^{-1}p(p + q)\\ 0 & -\rho & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 \end{bmatrix}, \end{equation} where $\alpha = p^2 + pq + q^2, \rho = \frac{p}{q}$. Therefore, $$\mathbb{P}^n = QJ^nQ^{-1}.$$ Under the constraint $p \in (0, 1), q \in (0, 1), \delta \in (0, 1), p + q + \delta = 1$, it follows that $$ |\delta \pm \sqrt{pq}| \leq \delta + \sqrt{pq} \leq \delta + \frac{1}{2}(p + q) < \delta + p + q = 1.$$ Hence $$\lim_{n \to \infty} J^n = \text{diag}(1, 0, 0, 0, 1).$$
Consequently,
$$\lim_{n \to \infty} \mathbb{P}^n = Q\text{diag}(1, 0, 0, 0, 1)Q^{-1} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ \alpha^{-1}q(p + q) & 0 & 0 & \alpha^{-1}p^2(1 - \beta) & \alpha^{-1}p^2\beta \\ \alpha^{-1}q^2 & 0 & 0 & \alpha^{-1}pq & \alpha^{-1}p^2 \\ 0 & 0 & 0 & 1 - \beta & \beta \\ 0 & 0 & 0 & 1 - \beta & \beta \end{bmatrix},$$ where $\beta = \frac{p}{p + q}$.