Rewrite the given absorbing stochastic matrix so that the absorbing states appear first, and identify the submatrices R and S.

Question

Rewrite the given absorbing stochastic matrix so that the absorbing states appear first, and identify the submatrices R and S. ​  ​

Quizplus · Accepted Answer

The absorbing states in the given stochastic matrix are the last two states (states 4 and 5). To rewrite the matrix with the absorbing states appearing first, we need to move the last two rows and columns to the beginning of the matrix. This gives us the following matrix:
[[1, 0, 0, 0, 0],
 [0, 1, 0, 0, 0],
 [0.6, 0, 0.3, 0.1, 0],
 [0.3, 0, 0.1, 0.5, 0.1],
 [0.1, 0, 0, 0.2, 0.7]]
The submatrix R corresponds to the transitions among the absorbing states, which is the 2x2 matrix in the top left corner:
[[1, 0],
 [0, 1]]
The submatrix S corresponds to the transitions from the transient states to the absorbing states, which is the 3x2 matrix in the top right corner:
[[0.6, 0],
 [0.3, 0],
 [0.1, 0]]

Rewrite the Given Absorbing Stochastic Matrix So That the Absorbing