Submatrices with the best-bounded inverses: revisiting the hypothesis

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 (\(n = 4, \, k = 2\)).