Limiting Matrices of Markov Chains with different submatrix dimensions.

Question

From what little I know of Markov chains that I got from this video,

The formula for calculating the fundamental matrix is $F = (I - Q)^{-1}$. However, how would one go about subtracting submatrices I and Q if they had different dimensions? In other words, if the number of absorbing states was different that the number of non-absorbing states.

Example:

[

    [1,   0,   0,   0,   0,   0],
    [0,   1,   0,   0,   0,   0],
    [0,   0,   1,   0,   0,   0],
    [0,   0,   0,   1,   0,   0],
    [0,   0,   0, 1/2,   0, 1/2],
    [0, 3/9, 2/9,   0, 4/9,   0]
]

Answer 1

In your example, we have 4 absorbing state and 2 transient. $$ Q = \begin{bmatrix} 0 & 1/2\\ 4/9 & 0 \end{bmatrix} $$ So $$ (I - Q) = \begin{bmatrix} 1 & -1/2\\ -4/9 & 1 \end{bmatrix} $$ and $$ (I - Q)^{-1} = \frac{9}{7} \begin{bmatrix} 1 & 1/2\\ 4/9 & 1 \end{bmatrix} $$