Hi, there

I have looked it up in the current textbook. The conventional numerical method to compute the inversion of an $n \times n$ matrix requires $O(n^3)$. However, for the following special matrix ${(E-a \cdot X)}^{-1}$,

where $E$ is an $n \times n$ identity matrix, $ a \in (0,1)$ is a scalar, and $X$ is a **sparse** stochastic matrix (the sum of each row is 1, and all its entries are between 0 and 1),

do you have some ideas to compute ${(E-a \cdot X)}^{-1}$ as fast as you can? (to reduce its $O(n^3)$ complexity is preferable, or approximate solution is also acceptable) ? Thanks in advance!