Approximating the inverse of a balanced symmetric matrix with positive elements

TL;DR

This paper introduces an approximation method for the inverse of balanced symmetric matrices with positive elements, providing explicit error bounds and demonstrating the approximation's accuracy improves with larger matrix size.

Contribution

It proposes a novel approximation formula for the inverse of such matrices and derives explicit uniform error bounds, enhancing computational efficiency.

Findings

The approximation is accurate to order 1/(n-1)^2.

Explicit bounds on the approximation error are derived.

The method is effective for large balanced symmetric matrices.

Abstract

For an $n\times n$ balanced symmetric matrix $T=(t_{i,j})$ with positive elements satisfying $t_{i,i}= \sum_{j\neq i} t_{i,j}$ and certain bounding conditions, we propose to use the matrix $S=(s_{i,j})$ to approximate its inverse, where $s_{i,j}=\delta_{i,j}/t_{i,i}-1/t_{..}$, $\delta_{i,j}$ is the Kronecker delta function, and $t_{..}=\sum_{i,j=1 }^{n}(1-\delta_{i,j}) t_{i,j}$. An explicit bound on the approximation error is obtained, showing that the inverse is well approximated to order $1/(n-1)^2$ uniformly.