Thread: Eigenvectors and transition matrix of markov chains.

1. Eigenvectors and transition matrix of markov chains.

Consider a Markov chain {$\displaystyle X_n : n = 0,1,2,...$} with states$\displaystyle S=(1,2)$ and transition matrix

$\displaystyle P= \begin{pmatrix} \frac{1}{4} & \frac{3}{4} \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix}$

Let $\displaystyle \mathbf{x}$ be a vector such that $\displaystyle \mathbf{x}P= \mathbf{x}$

where $\displaystyle x_1 > 0, x_2 > 0$ and $\displaystyle x_1 + x_2 = 1$

Let$\displaystyle \lambda$ and $\displaystyle \mathbf{y}$ satisfy

$\displaystyle \mathbf{y}P=\lambda \mathbf{y}$, where $\displaystyle |\lambda|<1$.

Let $\displaystyle \mathbf{\rho} = (\rho_1,\rho_2)$ be such that $\displaystyle \rho_1 > 0, \rho_2 > 0 , \rho_1+\rho_2=1$. Show that $\displaystyle \rho$ can be written as

$\displaystyle \mathbf{\rho}= \mathbf{x}+K \mathbf{y}$ for some constant $\displaystyle K$

I get:

$\displaystyle \mathbf{x}, \mathbf{y} \ are \ eigenvectors$, therefore
$\displaystyle \mathbf{\rho}=\alpha \mathbf{x}+ \beta \mathbf{y}$

However I cannot see how to make$\displaystyle \alpha = 1$

2. Re: Eigenvectors and transition matrix of markov chains.

The eigenvalues of $\displaystyle P$ are $\displaystyle \lambda_1=1,\lambda_2=-1/4$ . Using the definition of eigenvector, you'll find that necessarily $\displaystyle x=(2/5,3/5)$ and $\displaystyle y=(-\gamma,\gamma)\;\;(\gamma \neq 0)$ . Now, using $\displaystyle \rho_1+\rho_2=1$ it is easy to prove that $\displaystyle (\rho_1,\rho_2)=(2/5,3/5)+K(-\gamma,\gamma)$ for some $\displaystyle K$ .

3. Re: Eigenvectors and transition matrix of markov chains.

I am confused as to how you got

$\displaystyle y=(-\gamma, \gamma);\;(\gamma \neq 0)$

4. Re: Eigenvectors and transition matrix of markov chains.

Originally Posted by FGT12
I am confused as to how you got$\displaystyle y=(-\gamma, \gamma);\;(\gamma \neq 0)$
$\displaystyle yP=\lambda y$ implies $\displaystyle y$ is an eigenvector of $\displaystyle P$ associated to $\displaystyle \lambda$ . If $\displaystyle |\lambda|<1$ necessarily $\displaystyle \lambda=-1/4$ . Solve the system $\displaystyle y(P+(1/4)I)=0$ .