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在文献\cite{GTZ1997}中提出了以下假设：对于任意半正交$n \times k$矩阵，存在一个足够“好”的$k \times k$子矩阵。“好”意指其逆矩阵的谱范数有界。该假设指出，对于任意$k = 1, \ldots, n-1$，锐上界为$\sqrt{n}$。尽管有数值实验支持，该问题在所有非平凡情况（$1 < k < n-1$）下仍未解决。本文中，我们将给出其中最简单情形（$n = 4, \, k = 2$）的证明。