Supertropical Matrix Algebra II: Solving tropical equations

TL;DR

This paper advances supertropical matrix algebra by establishing the existence of a tangible adjoint, enabling unique solutions to supertropical equations and providing insights into eigenvector dependencies.

Contribution

It introduces a tangible adjoint for matrices over supertropical algebra, leading to a supertropical version of Cramer's rule and eigenvector analysis.

Findings

Existence of a tangible adjoint for supertropical matrices

Unique maximal solutions to supertropical vector equations

Example of matrices with dependent eigenvectors despite distinct eigenvalues

Abstract

We continue the study of matrices over a supertropical algebra, proving the existence of a tangible adjoint of $A$, which provides the unique right (resp. left) quasi-inverse maximal with respect to the right (resp. left) quasi-identity matrix corresponding to $A$; this provides a unique maximal (tangible) solution to supertropical vector equations, via a version of Cramer's rule. We also describe various properties of this tangible adjoint, and use it to compute supertropical eigenvectors, thereby producing an example in which an $n\times n$ matrix has $n$ distinct supertropical eigenvalues but their supertropical eigenvectors are tropically dependent.