On the submatrices with the best-bounded inverses

TL;DR

This paper proves a specific case of a hypothesis regarding submatrices with bounded inverses, confirming the existence of a submatrix with a certain singular value property for all n when k=2.

Contribution

The work provides a proof for the hypothesis when k=2 and arbitrary n, extending the known results beyond the previously verified small cases.

Findings

Confirmed the hypothesis for k=2 and arbitrary n.

Established the existence of a submatrix with a minimal singular value bound.

Extended the theoretical understanding of submatrix invertibility properties.

Abstract

The following hypothesis was formulated by Goreinov, Tyrtyshnikov, and Zamarashkin in \cite{goreinov1997theory}. If $U$ is $n\times k$ real matrix with the orthonormal columns $(n>k)$, then there exists a submatrix $Q$ of $U$ of size $k\times k$ such that its smallest singular value is at least $\frac{1}{\sqrt{n}}.$ Although this statement is supported by numerical experiments, the problem remains open for all $1<k<n-1,$ except for the case of $n = 4,\ k=2.$ In this work, we provide a proof for the case $k=2$ and arbitrary $n.$