Submatrices with the best-bounded inverses: an asymptotically tight upper bound for $\mathbb{C}^{n \times 2}$
This resolves the complex case of a long-standing conjecture, providing a theoretical guarantee for submatrix selection in numerical linear algebra.
The paper proves an asymptotically tight upper bound for the spectral norms of the best-bounded inverse 2×2 submatrices of any complex n×2 matrix with orthonormal columns, extending a result previously known only for real matrices.
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.