Dependence of supertropical eigenspaces

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 λ and corresponds to the columns of the eigenmatrix A+λI 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 (Formula presented.) and the connection of the independence question to generalized eigenvectors.