Simultaneous triangularization over max-algebras

TL;DR

This work addresses simultaneous triangularization of nonnegative matrices in max algebras using graph theory to translate algebraic questions into digraph acyclicity. It proves a sharp criterion: a matrix $A$ is triangularizable iff its digraph $G_A$ has no directed cycles of length $>1$, and extends this to pairs by showing $A,B$ are simultaneously triangularizable precisely when $G_A \cup G_B$ has no directed multi-vertex cycles (equivalently $G_{A \oplus B}$ is acyclic). The authors also investigate max commutators and commutants, establishing preservation of triangularizability under these operations in several cases, including when one matrix is unicellular, and link these results to nilpotency and projector structures. They introduce the tropical determinant based characteristic polynomial $P_{A,B}(z)$, proving that simultaneous triangularizability implies a product of $n$ linear factors and providing a necessary and sufficient condition for such factorization, with corollaries for diagonally dominant pairs and extensions to min-algebras, broadening the toolkit for max-algebraic linearization and related applications.

Abstract

The purpose of this article is to investigate triangularization and simultaneous triangularization of matrices over max algebras using graph theoretic methods. We establish a connection between commutators and commutants with simultaneous triangularization over max algebras. We also define the notion of characteristic polynomial of a collection in terms of the tropical determinant and determine when it can be written as a product of linear terms.

Theorems & Definitions (40)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Definition 2.4
• Theorem 2.5
• proof
• Example 2.6
• Definition 2.7
• Definition 3.1
• Theorem 3.3