Set Of Upper Triangular Matrices Research Articles

Ehrhart Positivity of Tesler Polytopes and Berline–Vergne’s Valuation

For {varvec{a}} in {mathbb {R}}_{ge 0}^{n}, the Tesler polytope {text {Tes}}_{n}({varvec{a}}) is the set of upper triangular matrices with non-negative entries whose hook sum vector is {varvec{a}}. Motivated by a conjecture of Morales, we study the questions of whether the coefficients of the Ehrhart polynomial of {text {Tes}}_n(1,1,dots ,1) are positive. We attack this problem by studying a certain function constructed by Berline–Vergne and its values on faces of a unimodularly equivalent copy of {text {Tes}}_n(1,1,dots ,1). We develop a method of obtaining the dot products appeared in formulas for computing Berline–Vergne’s function directly from facet normal vectors. Using this method together with known formulas, we are able to show Berline–Vergne’s function has positive values on codimension 2 and 3 faces of the polytopes we consider. As a consequence, we prove that the third and fourth coefficients of the Ehrhart polynomial of {text {Tes}}_{n}(1,dots ,1) are positive. Using the Reduction Theorem by Castillo and the second author, we generalize the above result to all deformations of {text {Tes}}_{n}(1,dots ,1) including all the integral Tesler polytopes.

Joint and Generalized Spectral Radius of Upper Triangular Matrices with Entries in a Unital Banach Algebra

In this paper, we discuss some properties of joint spectral {radius(jsr)} and generalized spectral radius(gsr) for a finite set of upper triangular matrices with entries in a Banach algebra and represent relation between geometric and joint/generalized spectral radius. Some of these are in scalar matrices, but some are different. For example for a bounded set of scalar matrices,$Sigma$, $r_*left(Sigmaright)= hat{r}left(Sigmaright)$, but for a bounded set of upper triangular matrices with entries in a Banach algebra($Sigma$), $r_*left(Sigmaright)neqhat{r}left(Sigmaright)$. We investigate when the set is defective or not and equivalent properties for having a norm equal to jsr, too.

The U ( n ) Gelfand–Zeitlin system as a tropical limit of Ginzburg–Weinstein diffeomorphisms

In this paper, we show that the Ginzburg–Weinstein diffeomorphism of Alekseev & Meinrenken (Alekseev, Meinrenken 2007 J. Differential Geom. 76 , 1–34. (10.4310/jdg/1180135664)) admits a scaling tropical limit on an open dense subset of . The target of the limit map is a product , where is the interior of a cone, T is a torus, and carries an integrable system with natural action-angle coordinates. The pull-back of these coordinates to recovers the Gelfand–Zeitlin integrable system of Guillemin & Sternberg (Guillemin, Sternberg 1983 J. Funct. Anal. 52 , 106–128. (10.1016/0022-1236(83)90092-7)). As a by-product of our proof, we show that the Lagrangian tori of the Flaschka–Ratiu integrable system on the set of upper triangular matrices meet the set of totally positive matrices for sufficiently large action coordinates. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

Jordan Triple Product Homomorphisms on Triangular Matrices to and from Dimension One

A map $\Phi$ is a Jordan triple product (JTP for short) homomorphism whenever $\Phi(A B A)= \Phi(A) \Phi(B) \Phi(A)$ for all $A,B$. We study JTP homomorphisms on the set of upper triangular matrices $\mathcal{T}_n(\mathbb{F})$, where $\Ff$ is the field of real or complex numbers. We characterize JTP homomorphisms $\Phi: \mathcal{T}_n(\mathbb{C}) \to \mathbb{C}$ and JTP homomorphisms $\Phi: \mathbb{F} \to \mathcal{T}_n(\mathbb{F})$. In the latter case we consider continuous maps and the implications of omitting the assumption of continuity.

Matrices that preserve the value of the generalized matrix function of the upper triangular matrices

Let H be a subgroup of the symmetric group S n and χ an irreducible character of H. In this paper we give conditions that characterize matrices that leave invariant the value of a given generalized matrix function on the set of upper triangular matrices. In some cases we describe completely these matrices.

Triangular partitioning for matrix inversion†

A triangular partitioning (additive) scheme for inverting a non-singular matrix is suggested. This scheme is direct and expresses the inverse matrix as a product of a lower triangular matrix and a set of upper triangular matrices