Submatrices with the best-bounded inverses: revisiting the hypothesis

TL;DR

The paper revisits the Goreinov–Tyrtyshnikov–Zamarashkin hypothesis on the existence, for any $n>k>0$ and any real $n\\times k$ matrix with orthonormal columns, of a $k\\times k$ submatrix whose inverse spectral norm is bounded by $\\sqrt{n}$, focusing on the simplest nontrivial case $n=4$, $k=2$. It develops a Plücker-coordinate framework, employs a thin CS-decomposition and Klein coordinates to reduce the problem to a pair of equations, and proves that the only feasible scaling under the bound is $X=Y=Z=1$, thereby establishing the tightness of the bound for this case and identifying $96$ equality subspaces with a concrete matrix. This work provides a rigorous proof for a specific small instance of the open problem and outlines a method potentially extendable to larger $n,k$ cases, with implications for pseudoskeleton-approximation theory.

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

Figures (1)

• Figure 1: The boundaries of the regions \ref{['eq:3']}. The contact points correspond to the subspaces the most deviating from the coordinate ones.