Proves an asymptotically tight upper bound on the spectral norm of the best-bounded-inverse 2x2 submatrix for arbitrary complex n x 2 orthonormal-column matrices.
On the submatrices with the best-bounded inverses
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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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math.NA 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
The equality criterion for submatrices with the best-bounded inverses is established for real n by 2 matrices.
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Submatrices with the best-bounded inverses: an asymptotically tight upper bound for $\mathbb{C}^{n \times 2}$
Proves an asymptotically tight upper bound on the spectral norm of the best-bounded-inverse 2x2 submatrix for arbitrary complex n x 2 orthonormal-column matrices.
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Submatrices with the best-bounded inverses: the equality criterion for $\mathbb{R}^{n \times 2}$
The equality criterion for submatrices with the best-bounded inverses is established for real n by 2 matrices.