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Submatrices with the best-bounded inverses: revisiting the hypothesis
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Submatrices with the best-bounded inverses: revisiting the hypothesis
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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 ($n = 4, \, k = 2$).
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On the submatrices with the best-bounded inverses
For k=2 and any n, every n x 2 orthonormal matrix U has a 2 x 2 submatrix Q with smallest singular value at least 1/sqrt(n).
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