Find the fundamental matrix F for the absorbing Markov chain, and find the product matrix FR.

Sir SagarDecember 16, 2014December 17, 20140

Find the fundamental matrix F for the absorbing Markov chain, and find the product matrix FR.

$\begin{bmatrix} 1 & 0 &0 \\ 0 & 1 & 0\\ 0.1& 0.1 & 0.8 \end{bmatrix}$

Solution

To find F, begin by rewriting the transition matrix in the form P = $\begin{bmatrix} I &O \\ R & Q \end{bmatrix}$ by switching rows if necessary to get the absorbing state in the upper left corner and the non-absorbing states in the lower right corner.

P = $\begin{bmatrix} 1 & 0 &0 \\ 0 & 1 & 0\\ 0.1& 0.1 & 0.8 \end{bmatrix}$

Thus ,Q = [0.8]and R = $\begin{bmatrix} 0.1 &0.1 \\ \end{bmatrix}$ .

Next , F = $(I_{n}-Q)^{-1}$ . Since Q is a 1 x 1 matrix, $I_{1}$ =[1]and so F = $([1]-[0.8])^{-1}$

= $([0.2])^{-1}$

= 5

Finally, find FR.

FR = [5] $\begin{bmatrix} 0.1 &0.1 \\ \end{bmatrix}$ .

$\begin{bmatrix} 0.5 &0.5 \\ \end{bmatrix}$ .

Leave a Reply Cancel reply