Find the fundamental matrix f for the absorbing Markov chain

Find the fundamental matrix f for the absorbing Markov chain, find the product mat [ ] F = [ ] FR = [ ]

Solution

To find F, begin by rewriting the transition matrix in the form ,

P =

| I O |

| R Q |

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=

| 1 0 0 |

| 0 1 0 |

|0.2 0.3 0.5|

Thus , Q=[0.5] and R = [ 0.2 0.3 ].

Next , F = (I_n - Q )^-1 . Since Q is a 1 x 1 matrix, I₁=[1]and so F = ([1] - [0.5)^-1 = 2

Finally, find FR,

FR = [2] [ 0.2 0.3 ] = [ 0.4 0.6 ] Answer