Characteristic polynomials of supertropical matrices

Supertropical matrix theory was investigated in [6], whose terminology we follow. In this work we investigate eigenvalues, characteristic polynomials and coefficients of characteristic polynomials of supertropical matrices and their powers, and obtain the analog to the basic property of matrices that any power of an eigenvalue of a matrix is an eigenvalue of the corresponding power of the matrix.Comment: Department of Mathematics, Bar-Ilan University, Ramat Gan 52900, Israel. Email: [email protected]. This paper is part of the author's Ph.D thesis, which was written at Bar-Ilan University under the supervision of Prof. L. H. Rowe

Tropical totally positive matrices

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.Comment: The first author has been partially supported by the PGMO Program of FMJH and EDF, and by the MALTHY Project of the ANR Program. The second author is sported by the French Chateaubriand grant and INRIA postdoctoral fellowshi

Dependence of Supertropical Eigenspaces

We study the pathology that causes tropical eigenspaces of distinct supertropical eigenvalues of a nonsingular 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 $\lambda$, and corresponds to the columns of the eigenmatrix $A+\lambda I$ 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

On pseudo-inverses of matrices and their characteristic polynomials in supertropical algebra

International audienceThe only invertible matrices in tropical algebra are diagonal matrices, permutation matrices and their products. However, the pseudo-inverse A ∇ , defined as 1 det(A) adj(A), with det(A) being the tropical permanent (also called the tropical determinant) of a matrix A, inherits some classical algebraic properties and has some surprising new ones. Defining B and B to be tropically similar if B = A ∇ BA, we examine the characteristic (max-)polynomials of tropically similar matrices as well as those of pseudo-inverses. Other miscellaneous results include a new proof of the identity for det(AB) and a connection to stabilization of the powers of definite matrices

Supertropical SLn

Also arXiv:1508.04483Extending earlier work on supertropical adjoints and applying symmetrization, we provide a symmetrized supertropical version SLn of the special linear group, which we partition into submonoids, based on " quasi-identity " matrices, and we display maximal sub-semigroups of SLn. We also study the monoid generated by SLn. Several illustrative examples are given of unexpected behavior. We describe the action of elementary matrices on SLn, which enables one to connect different matrices in SLn, but in a weaker sense than the classical situation

Factorization of Tropical Matrices

In contrast to the situation in classical linear algebra, not every tropically non-singular matrix can be factored into a product of tropical elementary matrices. We do prove the factorizability of any tropically non-singular 2x2 matrix and, relating to the existing Bruhat decomposition, determine which 3x3 matrices are factorizable. Nevertheless, there is a closure operation, obtained by means of the tropical adjoint, which is always factorizable, generalizing the decomposition of the closure operation * of a matrix.Comment: This paper is part of the author's PhD thesis, which was written at Bar-Ilan University under the supervision of Prof. L. H. Rowe

Brain volumetric changes in the general population following the COVID-19 outbreak and lockdown

The coronavirus disease 2019 (COVID-19) outbreak introduced unprecedented health-risks, as well as pressure on the economy, society, and psychological well-being due to the response to the outbreak. In a preregistered study, we hypothesized that the intense experience of the outbreak potentially induced stress-related brain modifications in the healthy population, not infected with the virus. We examined volumetric changes in 50 participants who underwent MRI scans before and after the COVID-19 outbreak and lockdown in Israel. Their scans were compared with those of 50 control participants who were scanned twice prior to the pandemic. Following COVID-19 outbreak and lockdown, the test group participants uniquely showed volumetric increases in bilateral amygdalae, putamen, and the anterior temporal cortices. Changes in the amygdalae diminished as time elapsed from lockdown relief, suggesting that the intense experience associated with the pandemic induced transient volumetric changes in brain regions commonly associated with stress and anxiety. The current work utilizes a rare opportunity for real-life natural experiment, showing evidence for brain plasticity following the COVID-19 global pandemic. These findings have broad implications, relevant both for the scientific community as well as the general public

Total non-negativity via valuations in tropical algebra

Tropical symposiumInternational audienc

Introduction to tropical total positivity

International audienc