On visualisation scaling, subeigenvectors and Kleene stars in max algebra

TL;DR

This paper explores the relationships between max algebra, convexity, and scaling problems, focusing on visualisation scaling, subeigenvectors, and Kleene stars in nonnegative matrices, with implications for matrix theory and convex geometry.

Contribution

It introduces new insights into visualisation scaling using max-algebraic subeigenvectors, Kleene stars, and convex geometric concepts, advancing understanding of matrix scaling in max algebra.

Findings

Characterization of visualisation scalings via max-algebraic subeigenvectors.

Connection between Kleene stars and convex geometry in matrix scaling.

Conditions for strict visualisation scaling in nonnegative matrices.

Abstract

The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is strict visualisation scaling, which means finding, for a given nonnegative matrix A, a diagonal matrix X such that all elements of X^{-1}AX are less than or equal to the maximum cycle geometric mean of A, with strict inequality for the entries which do not lie on critical cycles. In this paper such scalings are described by means of the max-algebraic subeigenvectors and Kleene stars of nonnegative matrices as well as by some concepts of convex geometry.