Dependence of tropical eigenspaces

Adi Niv 1, 2 Louis Rowen 3
1 TROPICAL - TROPICAL
Inria Saclay - Ile de France
2 MAXPLUS - Max-plus algebras and mathematics of decision
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France, Polytechnique - X, CNRS - Centre National de la Recherche Scientifique : UMR
Abstract : We study the pathology that causes tropical eigenspaces of distinct su-pertropical eigenvalues of a non-singular 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 adj(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 the " 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.
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Communications in Algebra, Taylor & Francis, 2017, 45 (3), pp. 924-942. 〈10.1080/00927872.2016.1172603〉
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Adi Niv, Louis Rowen. Dependence of tropical eigenspaces. Communications in Algebra, Taylor & Francis, 2017, 45 (3), pp. 924-942. 〈10.1080/00927872.2016.1172603〉. 〈hal-01253422〉

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