Submatrices with the best-bounded inverses: Studying $\mathds{R}^{n \times 2}$ and $\mathds{C}^{n \times 2}$

TL;DR

The paper investigates the maximal deviation, in terms of the largest principal angle, of a $2$-dimensional subspace from coordinate subspaces in both real and complex settings. It reframes the problem as a geometric optimization over isoperimetric polygons: real case via planar unit-perimeter polygons induced by the rows of an orthonormal $n\times 2$ matrix, and complex case via the Hopf map yielding spatial unit-perimeter polygons in $\mathbb{R}^3$. A real-case hypothesis ties the bound to $\max_{i,j} (|a_i|+|a_j|-|a_i+a_j|)$ with an equality structure, while the complex case leads to a bound $B_n^2$ and a tetrahedral extremal example showing a tighter bound than the real case; this is supported by explicit calculations for $n=4$. The approach provides a new geometric lens, connecting to Hausmann–Knutson’s isoperimetric polygon framework, and offers a clear pathway toward resolving the $2$-D GTZ1997 problem for general $n$.

Abstract

In both real and complex cases, we establish the connection of the problem about $2$-dimensional linear subspaces the most deviating from the coordinate ones with one simply formulated optimization problem for isoperimetric polygons in Euclidean spaces. This study thereby provides a new geometrical point of view on the $2$-dimensional case of the problem formulated by Goreinov, Tyrtyshnikov and Zamarashkin \cite{GTZ1997}, and at the same time presents a new application of the results by Hausmann and Knutson \cite{HK1997}.

Figures (1)

• Figure 1: Polygons described in Hypothesis \ref{['hr']}.