Submatrices with the best-bounded inverses: revisiting the hypothesis

Abstract: 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 ($n = 4, \, k = 2$).