Related papers: Tropical totally positive matrices
In this survey article, we describe recent work that connects three separate objects of interest: totally nonnegative matrices; quantum matrices; and matrix Poisson varieties.
We show that there is an isometry between the real ambient space of all Mueller matrices and the space of all Hermitian matrices which maps the Mueller matrices onto the positive semidefinite matrices. We use this to establish an optimality…
Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build…
Let $n \ge 2$ be an integer. In this note, we show that the {\it oriented} transition matrices over the field $\mathcal R$ of all real numbers (over the finite field $\mathcal Z_2$ of two elements respectively) of all continuous {\it vertex…
We study the homogeneous involutions on the full square matrices over an algebraically closed field endowed with a division grading with commutative support. We obtain the classification of the isomorphism and equivalence classes for the…
In this paper, we consider matrices whose entries are combinatorial sequences which can be expressed in terms of a convolution of elementary and complete homogeneous symmetric functions. We establish the total positivity of these matrices…
Kraus maps (completely positive trace preserving maps) arise classically in quantum information, as they describe the evolution of noncommutative probability measures. We introduce tropical analogues of Kraus maps, obtained by replacing the…
We exhibit a lower-triangular matrix of polynomials $T(a,c,d,e,f,g)$ in six indeterminates that appears empirically to be coefficientwise totally positive, and which includes as a special case the Eulerian triangle. We prove the…
We study three combinatorial models for the lower-triangular matrix with entries $t_{n,k} = \binom{n}{k} n^{n-k}$: two involving rooted trees on the vertex set $[n+1]$, and one involving partial functional digraphs on the vertex set $[n]$.…
We define tropical analogues of the notions of linear space and Plucker coordinate and study their combinatorics. We introduce tropical analogues of intersection and dualization and define a tropical linear space built by repeated…
Using the interpretation of the ultradiscretization procedure as a non-Archimedean valuation, we use results of tropical geometry to show how roots and poles manifest themselves in piece-wise linear systems as points of…
We introduce a faithful tropical linear representation of the Chinese monoid, and thus prove that this monoid admits all the semigroup identities satisfied by tropical triangular matrices.
An $n\times n$ symmetric matrix $A$ is copositive if the quadratic form $x^TAx$ is nonnegative on the nonnegative orthant. The cone of copositive matrices strictly contains the cone of completely positive matrices, i.e., all matrices of the…
We investigate the Voronoi diagrams with respect to an asymmetric tropical distance function, also for infinite point sets. These turn out to be much better behaved than the tropical Voronoi diagrams arising from the standard tropical…
For a class of maximally degenerate families of Calabi-Yau hypersurfaces of complex projective space, we study associated non-Archimedean and tropical Monge-Amp\`ere equations, taking place on the associated Berkovich space, and the…
We define ''convergence'' for noncommutative power series and construct two topologies on the algebra of power series, convergent with respect to a positive radius. We indicate all finite dimensional continuous representations of this…
We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and a generalization of the Sturmfels-Tevelev…
A real symmetric matrix $M$ is completely positive semidefinite if it admits a Gram representation by (Hermitian) positive semidefinite matrices of any size $d$. The smallest such $d$ is called the (complex) completely positive semidefinite…
The only invertible matrices in tropical algebra are diagonal matrices, permutation matrices and their products. However, the pseudo-inverse $A^\nabla$, defined as $\frac{adj(A)}{det(A)}$, with $det(A)$ being the tropical permanent (also…
We show that the moduli spaces of irreducible labeled parametrized marked rational curves in toric varieties can be embedded into algebraic tori such that their tropicalizations are the analogous tropical moduli spaces. These embeddings are…