Approximating the inverse of a balanced symmetric matrix with positive elements

Abstract

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

Major revisions, 8 pages