Tight bounds of norms of Wasserstein metric matrix

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.