Submatrices with the best-bounded inverses: an asymptotically tight upper bound for $\mathbb{C}^{n \times 2}$

TL;DR

This paper extends a recent real matrix inverse bound result to complex matrices, establishing an asymptotically tight upper bound for spectral norms of best-bounded inverses in complex $n imes 2$ matrices.

Contribution

It proves an asymptotically tight upper bound for spectral norms of inverse submatrices in complex matrices, generalizing previous real matrix results.

Findings

Established an asymptotically tight upper bound for complex matrices.

Extended the real matrix inverse bound to complex matrices.

Provided theoretical proof for the spectral norm bounds.

Abstract

The long-standing hypothesis formulated by Goreinov, Tyrtyshnikov and Zamarashkin \cite{GTZ1997} has recently been solved affirmatively in the case of real two-column matrices by Sengupta and Pautov \cite{SP2026}. In this paper, we consider the complex variant of this problem and prove the asymptotically tight upper bound for spectral norms of the best-bounded inverse $2 \times 2$ submatrices of an arbitrary complex $n \times 2$ matrix with orthonormal columns.