Tight bounds of norms of Wasserstein metric matrix

TL;DR

This paper investigates algebraic and computational properties of Wasserstein-1 metric matrices, providing sharper bounds on their norms, inverses, and condition numbers, along with eigenvalue and numerical range analyses.

Contribution

It introduces new bounds on matrix norms and condition numbers, and offers novel decompositions of Wasserstein matrices, advancing understanding of their algebraic structure.

Findings

Sharper upper bounds on $1$ and $inity$-norms of Wasserstein matrices.

New decompositions of Wasserstein matrices and their inverses.

Regions for eigenvalues and numerical ranges of these matrices.

Abstract

Very recently, Bai [Linear Algebra Appl., 681:150-186, 2024 \& Appl. Math. Lett., 166:109510, 2025] studied some concrete structures, and obtained essential algebraic and computational properties of the one-dimensional, two-dimensional and generalized Wasserstein-1 metric matrices. This article further studies some algebraic and computational properties of these classes of matrices. Specifically, it provides lower and upper bounds on the $1,2,\infty$-norms of these matrices, their inverses, and their condition numbers. For the $1$ and $\infty$-norms, these upper bounds are much sharper than the existing ones established in the above-mentioned articles. These results are also illustrated using graphs, and the computation of the bounds is presented in tables for various matrix sizes. It also finds regions for the inclusion of the eigenvalues and numerical ranges of these matrices. A new decomposition of the Wasserstein matrix in terms of the Hadamard product can also be seen. Also, a decomposition of the Hadamard inverse of the Wasserstein matrix is obtained. Finally, a few bounds on the condition number are obtained using numerical radius inequalities.