1,225 research outputs found
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
A Self-learning Algebraic Multigrid Method for Extremal Singular Triplets and Eigenpairs
A self-learning algebraic multigrid method for dominant and minimal singular
triplets and eigenpairs is described. The method consists of two multilevel
phases. In the first, multiplicative phase (setup phase), tentative singular
triplets are calculated along with a multigrid hierarchy of interpolation
operators that approximately fit the tentative singular vectors in a collective
and self-learning manner, using multiplicative update formulas. In the second,
additive phase (solve phase), the tentative singular triplets are improved up
to the desired accuracy by using an additive correction scheme with fixed
interpolation operators, combined with a Ritz update. A suitable generalization
of the singular value decomposition is formulated that applies to the coarse
levels of the multilevel cycles. The proposed algorithm combines and extends
two existing multigrid approaches for symmetric positive definite eigenvalue
problems to the case of dominant and minimal singular triplets. Numerical tests
on model problems from different areas show that the algorithm converges to
high accuracy in a modest number of iterations, and is flexible enough to deal
with a variety of problems due to its self-learning properties.Comment: 29 page
M-tensors and The Positive Definiteness of a Multivariate Form
We study M-tensors and various properties of M-tensors are given. Specially,
we show that the smallest real eigenvalue of M-tensor is positive corresponding
to a nonnegative eigenvector. We propose an algorithm to find the smallest
positive eigenvalue and then apply the property to study the positive
definiteness of a multivariate form
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