178 research outputs found
Supertropical matrix algebra III: Powers of matrices and generalized eigenspaces
We investigate powers of supertropical matrices, with special attention to
the role of the coefficients of the supertropical characteristic polynomial
(especially the supertropical trace) in controlling the rank of a power of a
matrix. This leads to a Jordan-type decomposition of supertropical matrices,
together with a generalized eigenspace decomposition of a power of an arbitrary
supertropical matrix.Comment: 21 page
Dependence of Supertropical Eigenspaces
We study the pathology that causes tropical eigenspaces of distinct
supertropical eigenvalues of a nonsingular matrix , 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
, and corresponds to the columns of the eigenmatrix 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.Comment: The first author is sported by the French Chateaubriand grant and
INRIA postdoctoral fellowshi
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