Submatrices with the best-bounded inverses: revisiting the hypothesis
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914922966810624 |
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| author | Nesterenko, Yuri |
| author_facet | Nesterenko, Yuri |
| contents | The following hypothesis was put forward by Goreinov, Tyrtyshnikov and Zamarashkin in \cite{GTZ1997}. For arbitrary real $n \times k$ matrix with orthonormal columns a sufficiently "good" $k \times k$ submatrix exists. "Good" in the sense of having a bounded spectral norm of its inverse. The hypothesis says that for arbitrary $k = 1, \ldots, n-1$ the upper bound can be set at $\sqrt{n}$. Supported by numerical experiments, the problem remained open for all non-trivial cases ($1 < k < n-1$). In this paper we will give the proof for the simplest of them ($n = 4, \, k = 2$). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_07492 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Submatrices with the best-bounded inverses: revisiting the hypothesis Nesterenko, Yuri Numerical Analysis The following hypothesis was put forward by Goreinov, Tyrtyshnikov and Zamarashkin in \cite{GTZ1997}. For arbitrary real $n \times k$ matrix with orthonormal columns a sufficiently "good" $k \times k$ submatrix exists. "Good" in the sense of having a bounded spectral norm of its inverse. The hypothesis says that for arbitrary $k = 1, \ldots, n-1$ the upper bound can be set at $\sqrt{n}$. Supported by numerical experiments, the problem remained open for all non-trivial cases ($1 < k < n-1$). In this paper we will give the proof for the simplest of them ($n = 4, \, k = 2$). |
| title | Submatrices with the best-bounded inverses: revisiting the hypothesis |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2303.07492 |