A Note on Equivalent Conditions for Majorization

TL;DR

The paper develops two novel characterizations of majorization: $x \prec y$ iff there exists a lower triangular row-stochastic matrix $\mathbf{L}$ with $\mathbf{y} = \mathbf{x} \mathbf{L}$, and $x = y \mathbf{U}$ iff there exists an upper triangular row-stochastic matrix $\mathbf{U}$ with $x = \mathbf{y} \mathbf{U}$, each realized as a product of at most $n-1$ elementary transforms ($\mathbf{A}$-transforms for the lower-triangular case and $\mathbf{B}$-transforms for the upper-triangular case). These decompositions yield sparse, structured matrices (at most $2n-1$ nonzeros in the accumulated upper-triangular case) and provide practical pathways to study Schur-concave properties. The authors leverage the $\mathbf{B}$-transform decomposition to derive an improved entropy inequality on the probability simplex $\mathcal{P}_n$, linking majorization, the Dobrushin coefficient $\alpha(\mathbf{U})$, and the distributional term $\sum_i x_i \log_2 1/(\mathbf{u}\mathbf{U})_i$. Overall, the work extends classical majorization theory with transform-based characterizations and demonstrates tangible implications for entropy bounds in information-theoretic settings.

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.

Theorems & Definitions (34)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.1
• Theorem 3.1
• proof
• Corollary 3.2
• proof
• Lemma 3.3
• proof