Stochastic matrices and majorization in max algebra

TL;DR

This work develops a framework for majorization in max algebra by introducing max-doubly stochastic matrices and analyzing their structure and extremal points. It proves that $x \prec_{\max} y$ holds exactly when there exists a max-doubly stochastic matrix $D$ with $x = D \otimes y$, equivalently tying majorization to the equality of maxima and a lower bound on minima. The authors show that the max-doubly stochastic set has intricate extreme-point behavior and that Birkhoff's theorem fails in this tropical setting, while also establishing a preorder structure for max-majorization and a Minkowski-like description of the majorized set. Overall, the paper extends majorization concepts to tropical/max-algebraic contexts and clarifies how vector relations behave under max-doubly stochastic transformations, with implications for tropical linear systems and inequalities.

Abstract

In this paper, we introduce and characterize max-doubly stochastic matrices within the framework of max algebra, where the operations are defined as $x \oplus y = \max(x, y)$ and $x \otimes y = xy$. We explore the fundamental properties of max-doubly stochastic matrices and their role in vector majorization. Specifically, we establish that for vectors $x$ and $y$ in max algebra, $x$ is majorized by $y$ if there exists a max-doubly stochastic matrix $D$ such that $x = D \otimes y$. This provides a new approach to majorization theory within tropical mathematics and enhances the understanding of vector relations in max algebra.

Figures (4)

• Figure 1: Set of $S$
• Figure 2: Set of $\hbox{ext}_{\max}\left( S \right)$
• Figure 3: $\left\{ x\in\mathbb R _{\max}^{2} |~ x\prec_{\max} \left( 2,1\right)^{T} \right\}$
• Figure 4: $\left\{ x\in\mathbb R _{\max}^{3} |~ x\prec_{\max} \left( 1,2,3\right)^{T} \right\}$

Theorems & Definitions (13)

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