# Approximating the inverse of a diagonally dominant matrix with positive elements

## TL;DR

This paper proposes a simple diagonal matrix approximation for the inverse of diagonally dominant matrices with positive elements, providing explicit error bounds that demonstrate high accuracy.

## Contribution

The paper introduces a diagonal approximation method for matrix inversion and derives explicit error bounds, improving understanding of approximation accuracy for such matrices.

## Key findings

- The approximation error is of order O(n^{-2}).
- The diagonal matrix S closely approximates T^{-1}.
- The method is effective for diagonally dominant matrices with positive elements.

## Abstract

For an $n\times n$ diagonally dominant matrix $T=(t_{i,j})_{n\times n}$ with positive elements satisfying certain bounding conditions, we propose to use a diagonal 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 delta function. We derive an explicitly upper bound on the approximation error, which is in the magnitude of $O(n^{-2})$. It shows that $S$ is a very good approximation to $T^{-1}$.