Markov Chain problem.

• March 18th 2011, 03:25 AM
harrybarryt
Markov Chain problem.
With two-state Markov chain which a one-step transition matrix :

P = [ 7/8 1/8 ]
[ 1/8 7/8 ]

How would i find the eigenvalues of P and derive an explicit expression for Pn?.(Rock)
• March 18th 2011, 03:40 AM
FernandoRevilla
Quote:

Originally Posted by harrybarryt
How would i find the eigenvalues of P and derive an explicit expression for Pn?.

The eigenvalues of $P$ are $\lambda_1=1,\;\lambda_2=3/4$ and the corresponding eigenvectors

$v_1=(1,1),\;v_2=(-1,1)$

so,

$P^r=\begin{bmatrix}{1}&{-1}\\{1}&{\;\;\;1}\end{bmatrix}\begin{bmatrix}{1^r} &{0}\\{0}&{(3/4)^r}\end{bmatrix}\begin{bmatrix}{1}&{-1}\\{1}&{\;\;\;1}\end{bmatrix}^{-1}=\ldots$