A Note on Equivalent Conditions for Majorization

TL;DR

This paper presents new characterizations of majorization using triangular stochastic matrices and linear transforms, leading to an improved entropy inequality, enhancing understanding of the concept's properties.

Contribution

The paper introduces novel characterizations of majorization through triangular stochastic matrices and linear transforms, providing new tools for analysis.

Findings

New characterizations of majorization using upper and lower triangular stochastic matrices.

Derived an improved entropy inequality based on these characterizations.

Enhanced theoretical understanding of majorization properties.

Abstract

In this paper, we introduce novel characterizations of the classical concept of majorization in terms of upper triangular (resp., lower triangular) row-stochastic matrices, and in terms of sequences of linear transforms on vectors. We used our new characterizations of majorization to derive an improved entropy inequality.