  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.
