Dependence of Supertropical Eigenspaces

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.