Dependence of Supertropical Eigenspaces

TL;DR

This paper investigates why supertropical eigenvectors of matrices become dependent in higher dimensions and identifies conditions under which their independence is restored, enhancing understanding of supertropical eigenstructure.

Contribution

It analyzes the dependence pathology of supertropical eigenspaces, introduces a difference criterion for independence, and explores implications for generalized eigenvectors.

Findings

Eigenvectors are independent in low dimensions.

A difference criterion ensures eigenvector independence in higher dimensions.

Connections to generalized eigenvectors are established.

Abstract

We study the pathology that causes tropical eigenspaces of distinct supertropical eigenvalues of a nonsingular matrix $A$, to be dependent. We show that in lower dimensions the eigenvectors of distinct eigenvalues are independent, as desired. The index set that differentiates between subsequent essential monomials of the characteristic polynomial, yields an eigenvalue $\lambda$, and corresponds to the columns of the eigenmatrix $A+\lambda I$ from which the eigenvectors are taken. We ascertain the cause for failure in higher dimensions, and prove that independence of the eigenvectors is recovered in case a certain "difference criterion" holds, defined in terms of disjoint differences between index sets of subsequent coefficients. We conclude by considering the eigenvectors of the matrix $A^\nabla : = \det(A)^{-1}\adj(A)$ and the connection of the independence question to generalized eigenvectors.