On tropical Kleene star matrices and alcoved polytopes

Abstract

In this paper we give a short, elementary proof of a known result in tropical mathematics, by which the convexity of the column span of a zero--diagonal real matrix $A$ is characterized by $A$ being a Kleene star. We give applications to alcoved polytopes, using normal idempotent matrices (which form a subclass of Kleene stars). For a normal matrix we define a norm and show that this is the radius of a hyperplane section of its tropical span.

arXiv commentThis is a newer revised version. Four figures included. To appear in Kybernetika

References in corpus (6)

• Duality and separation theorems in idempotent semimodules
• Generators, extremals and bases of max cones
• On visualisation scaling, subeigenvectors and Kleene stars in max algebra
• Computing the vertices of tropical polyhedra using directed hypergraphs
• Max-plus definite matrix closures and their eigenspaces
• Six combinatorial clases of maximal convex tropical polyhedra

Cited by in corpus (3)

• Tropical implementation of the Analytical Hierarchy Process decision method
• Quasi--Euclidean classification of Alcoved Convex Polyhedra
• Multivariate volume, Ehrhart, and $h^{\ast }$-polynomials of polytropes