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Tropical totally positive matrices

Stéphane Gaubert 1, 2 Adi Niv 1, 2
2 TROPICAL - TROPICAL
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : We investigate the tropical analogues of totally positive and totally nonnegative matrices. These arise when considering the images by the nonarchimedean valuation of the corresponding classes of matrices over a real nonarchimedean valued field, like the field of real Puiseux series. We show that the nonarchimedean valuation sends the totally positive matrices precisely to the Monge matrices. This leads to explicit polyhedral representations of the tropical analogues of totally positive and totally nonnegative matrices. We also show that sign-nonsingular tropical totally nonnegative matrix can be factorized in terms of elementary matrices. We finally determine the eigenvalues of tropical totally positive matrices, and relate them with the eigenvalues of totally positive matrices over nonarchimedean fields.
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https://hal.inria.fr/hal-01423747
Contributor : Stephane Gaubert <>
Submitted on : Saturday, December 31, 2016 - 1:22:05 PM
Last modification on : Friday, April 30, 2021 - 9:53:58 AM

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Stéphane Gaubert, Adi Niv. Tropical totally positive matrices. Journal of Algebra, Elsevier, 2018, 515, pp.511-544. ⟨10.1016/j.jalgebra.2018.07.005⟩. ⟨hal-01423747⟩

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