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For an $n\times n$ diagonally dominant matrix $T=(t_{i,j})_{n\times n}$ withpositive elements satisfying certain bounding conditions, we propose to use adiagonal matrix $S=(s_{i,j})_{n\times n}$ to approximate the inverse of $T$,where $s_{i,j}=\delta_{i,j}/t_{i,i}$ and $\delta_{i,j}$ is the Kronecker deltafunction. We derive an explicitly upper bound on the approximation error, whichis in the magnitude of $O(n^{-2})$. It shows that $S$ is a very goodapproximation to $T^{-1}$.

Approximating the inverse of a diagonally dominant matrix with positive elements