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 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 A ∇ := 1 det(A) adj(A) and the connection of the independence question to generalized eigenvectors.

Independence and orthogonality of algebraic eigenvectors over the max-plus algebra

“…This means that the column rank of the n × r matrix consisting of r algebraic eigenvectors is r. In Section 3, we discuss the independence of algebraic eigenvectors in the latter sense. Note that the independence of supertropical eigenvectors is discussed in the former sense [24]. We first prove that the algebraic eigenspaces with respect to two distinct algebraic eigenvalues have no intersection except for the zero vector.…”

“…In this section, we discuss the max-plus analogue of this fact. First, we see an example taken from [24], which is written in the supertropical settings.…”

On the vectors associated with the roots of max-plus characteristic polynomials

“…Hence, it seems natural to define the analogues of eigenvectors by using tropical geometry. In fact, there is an approach from the perspective of supertropical algebra [9], [10], [11], [14] in line with this idea, but it would produce more "eigenvectors" than expected, that is, the number of independent eigenvectors could exceed the multiplicity of the root. The computation of one eigenvector is described in [10], but finding all eigenvectors is difficult.…”