Submatrices with the best-bounded inverses: the equality criterion for $\mathbb{R}^{n \times 2}$

TL;DR

This paper addresses the problem of identifying submatrices with optimal inverse bounds in two-column matrices, providing an equality criterion that complements recent positive solutions to a longstanding hypothesis.

Contribution

It introduces an equality criterion for submatrices with the best-bounded inverses in the specific case of matrices with two columns, extending prior theoretical results.

Findings

Confirmed the positive solution to the hypothesis for two-column matrices.

Provided an equality criterion to identify submatrices with optimal inverse bounds.

Extended the theoretical understanding of matrix inverse bounds in this context.

Abstract

The long-standing hypothesis formulated by Goreinov, Tyrtyshnikov and Zamarashkin \cite{GTZ1997} has recently been solved positively by Sengupta and Pautov \cite{SP2026} in the case of two-column matrices. In this paper, we complement their elegant proof with the equality criterion.