The following hypothesis was put forward by Goreinov, Tyrtyshnikov and Zamarashkin in \cite{GTZ1997}. For arbitrary semi-orthogonal $n \times k$ matrix 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 sharp upper bound is $\sqrt{n}$. Supported by numerical experiments, the problem remains open for all non-trivial cases ($1 < k < n-1$). In this paper, we will give the proof for the simplest of them ($n = 4, \, k = 2$).
翻译:Goreinov、Tyrtyshnikov和Zamarashkin在\cite{GTZ1997}中提出了以下假设。对于任意的半正反方$\time k$ k$ 矩阵来说,一个足够“好” $k\time k$ 子矩阵存在。“好” 是指其反面的光谱规范。假设说,对于任意的美元=1,\ldots, n-1美元,尖尖的上限是$\sqrt{n}。在数字实验的支持下,所有非三审案件的问题仍然存在(1美元 < k < n-1美元) 。在本文中,我们将为最简单的案件提供证据(美元=4, k=2美元)。</s>