Question

Transition Matrix - Markov Chains

Let \[P _{2}=\left[\begin{matrix}1-\alpha & \alpha \\ \beta & 1-\beta\end{matrix}\right]\]
Calculate it's n-step transition matrix \[(P_2)^n\]
I know I need to start by finding the eigenvalues:
So we set \[ \det(P - \lambda I)=0\]
and I get that \[\alpha+\beta-1=0\]
but how do I interpret my eigenvalues/vectors from this?

Answer 1

how do you get that value? and what are your eiven values?

Answer 2

I think I've just figured out exactly what I've done wrong!

Answer 3

okay .. that's great!!

Answer 4

I didn't actually use the formula, I just calculated the determinant and set it to zero!

Answer 5

Actually no I haven't worked it out, so if I take the determinant and set it to zero, I get:
\[(1-\alpha-\lambda)(1-\beta-\lambda)-\alpha \beta=0\] to which I got: \[\lambda^2+(\alpha+\beta-2)\lambda-(\alpha+\beta-1)=0\]
So if:\[\alpha + \beta=0 \quad \text{we obtain an eigenvalue of 1}\] \[\text{if} \quad \alpha+\beta>0 \quad\text{then we obtain an eigenvalue of} \quad 1-\alpha-\beta\]

How do I then calculate my eigenvectors?

Answer 6

find your eiven values first ... use that eigen value in
\[ P_2 X = \lambda X \]
you will get two set of equations by comparing the rows of X. solve it and can you have your eigen vectors.