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arXiv · 2604.05944

On the submatrices with the best-bounded inverses

Richik Sengupta·Mikhail Pautov
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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 $n.$

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Richik Sengupta, Mikhail Pautov. 2026-04-16. On the submatrices with the best-bounded inverses. https://arxiv.org/abs/2604.05944

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