On the Frobenius norm of the inverse of a non-negative matrix

TL;DR

This paper establishes a new lower bound for the Frobenius norm of the inverse of non-negative matrices, resolving the S-matrix conjecture for all sufficiently large dimensions.

Contribution

It provides a modest but crucial improvement in the lower bound, enabling the full resolution of the S-matrix conjecture for large dimensions.

Findings

New lower bound for Frobenius norm of inverse matrices

Resolution of the S-matrix conjecture for large dimensions

Improved understanding of non-negative matrix inverses

Abstract

We prove a new lower bound for the Frobenius norm of the inverse of an non-negative matrix. This bound is only a modest improvement over previous results, but is sufficient for fully resolving a conjecture of Harwitz and Sloane, commonly referred to as the S-matrix conjecture, for all dimensions larger than a small constant.