# Tropical totally positive matrices

• Published in 2016
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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 tropical totally nonnegative matrices with a finite permanent can be factorized in terms of elementary matrices. We finally determine the eigenvalues of tropical totally nonnegative matrices, and relate them with the eigenvalues of totally nonnegative matrices over nonarchimedean fields.

### BibTeX entry

@article{Tropicaltotallypositivematrices,
title = {Tropical totally positive matrices},
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 tropical totally nonnegative matrices with a finite permanent
can be factorized in terms of elementary matrices. We finally determine the
eigenvalues of tropical totally nonnegative matrices, and relate them with the
eigenvalues of totally nonnegative matrices over nonarchimedean fields.},
url = {http://arxiv.org/abs/1606.00238v3 http://arxiv.org/pdf/1606.00238v3},
author = {St{\'{e}}phane Gaubert and Adi Niv},
comment = {},
urldate = {2017-05-02},
archivePrefix = {arXiv},
eprint = {1606.00238},
primaryClass = {math.AC},
year = 2016,
collections = {Attention-grabbing titles,Unusual arithmetic}
}