I have recently started learning Markov Chains and feel somewhat out of my depth, as im not a mathematics student. I am trying to calculate $p^n_{11}$ for a 3 state chain with the following transition matrix and eigenvalues: $$ \begin{bmatrix} 1/2 & 1/3 & 1/6 \\ 0 & 1/3 & 2/3 \\ 1/2 & 0 & 1/2 \\ \end{bmatrix} $$
1 , $\frac{1 + \sqrt{5}i}{6}$, $\frac{1 - \sqrt{5}i}{6}$
You can represent $p^n_{11}$ as:
$p^n_{11} = C_1(1)^n + C_2(\frac{1 + \sqrt{5}i}{6})^n + C_3(\frac{1 - \sqrt{5}i}{6})^n$
Also:
$(\frac{1 \pm \sqrt{5}i}{6})^n = (\frac{1}{6})^nr^n[\beta cos(n\theta) + \gamma sin(n\theta)]$
Where $r$ is the modulus and $\theta$ is the argument of the complex number.
$p^n_{11} = \alpha + (\frac{1}{6})^n(\sqrt{6})^n[\beta cos(n*arctan(\sqrt{5})) + \gamma sin(n*arctan(\sqrt{5}))]$
This was tractable and simple to solve when the complex nuumber was something like $\frac{i}{2}$. With this problem, you get the argument as $arctan(\sqrt{5})$ which makes it difficult to solve the system of equations you get when you input the initial conditions. The system of equations is:
$\alpha + \beta = 1 $ since $p^0_{11} = 1$
$\alpha + \frac{\beta}{6} + \frac{\gamma \sqrt{5}}{6} = 1/2 $ since $p^1_{11} = 1/2$
For $n=2$, you get $sin(2*arctan(\sqrt{5})) \approx 0.74535...$ which I evaluated using an online calculator. This becomes difficult to solve the equations. I am stuck here. Could you please help me out? Thanks!