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

TL;DR

This paper explores the geometric properties of 2-dimensional subspaces in real and complex spaces, linking them to an optimization problem related to isoperimetric polygons, offering new insights into matrix inverse bounds.

Contribution

It introduces a novel geometric perspective on the 2D case of a matrix inverse problem, connecting it with isoperimetric polygons and existing mathematical results.

Findings

Established a connection between subspace deviation and isoperimetric polygons.

Provided a new geometric interpretation of the inverse problem.

Linked the problem to existing results by Hausmann and Knutson.

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}.