Elementary equivalence of the semigroup of invertible matrices with nonnegative elements

Generalized Projection Operators in Geometric Algebra

Given an automorphism and an anti-automorphism of a semigroup of a Geometric Algebra, then for each element of the semigroup a (generalized) projection operator exists that is defined on the entire Geometric Algebra. A single fundamental theorem holds for all (generalized) projection operators. This theorem makes previous projection operator formulas equivalent to each other. The class of generalized projection operators includes the familiar subspace projection operation by implementing the automorphism `grade involution' and the anti-automorphism `inverse' on the semigroup of invertible vers…

Multiplication of matrices over lattices

We study the multiplication operation of square matrices over lattices. If the underlying lattice is distributive, then matrices form a semigroup; we investigate idempotent and nilpotent elements and the maximal subgroups of this matrix semigroup. We prove that matrix multiplication over nondistributive lattices is antiassociative, and we determine the invertible matrices in the case when the least or the greatest element of the lattice is irreducible.

半正矩阵的可逆线性保持映射——一种无维度方法

Invertible linear preservers of semipositive matrices - a dimension free approach

本文解决了半正矩阵可逆线性保持映射猜想中m<n的情形，其证明适用于任意正整数m、n，核心是对半正矩阵集合中最大维度仿射子空间分类。

Factorizations of $k$-Nonnegative Matrices

A matrix is $k$-nonnegative if all its minors of size $k$ or less are nonnegative. We give a parametrized set of generators and relations for the semigroup of $k$-nonnegative $n\times n$ invertible matrices in two special cases: when $k = n-1$ and when $k = n-2$, restricted to unitriangular matrices. For these two cases, we prove that the set of $k$-nonnegative matrices can be partitioned into cells based on their factorizations into generators, generalizing the notion of Bruhat cells from totally nonnegative matrices. Like Bruhat cells, these cells are homeomorphic to open balls and have a to…

Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem

Call an $n$-by-$n$ invertible matrix $S$ a \emph{Perron similarity} if there is a real non-scalar diagonal matrix $D$ such that $S D S^{-1}$ is entrywise nonnegative. We give two characterizations of Perron similarities and study the polyhedra $\mathcal{C}(S) := \{ x \in \mathbb{R}^n: S D_x S^{-1} \geq 0,~D_x := \text{diag}(x) \}$ and $\mathcal{P})(S) := \{x \in \mathcal{C}(S) : x_1 = 1 \}$, which we call the \emph{Perron spectracone} and \emph{Perron spectratope}, respectively. The set of all normalized real spectra of diagonalizable nonnegative matrices may be covered by Perron spectratopes,…

Cholesky decomposition of positive semidefinite matrices over commutative semirings

We prove that over a commutative semiring every symmetric strongly invertible matrix with nonnegative numerical range has a Cholesky decomposition.

无限可分矩阵的线性保持算子

Linear Preservers of Infinitely Divisible Matrices

本文刻画了实数矩阵空间上保持强无限可分矩阵和无限可分非负矩阵的双射线性映射，分别采用Zariski稠密技术和锥保持方法。

Kronecker products of Perron similarities

An invertible matrix is called a Perron similarity if one of its columns and the corresponding row of its inverse are both nonnegative or both nonpositive. Such matrices are of relevance and import in the study of the nonnegative inverse eigenvalue problem. In this work, Kronecker products of Perron similarities are examined and used to to construct ideal Perron similarities all of whose rows are extremal.

广义马歇尔商与连续可微函数的实半群

Generalized Marshall Quotients and Real Semigroups of Continuous and Differentiable Functions

本文针对实半群理论在连续函数环应用中的拓扑约束问题，引入广义马歇尔商，给出实半群新示例，得出商的可逆元群构成实约化超域，解决相关开放问题，并应用超代数机制建立广义不等式。

The automorphisms group and the classification of gradings of finite dimensional associative algebras

Let $A$ be an $n$-dimensional algebra over a field $k$ and $a(A)$ its quantum symmetry semigroup. We prove that the automorphisms group ${\rm Aut}_{\rm Alg} (A)$ of $A$ is isomorphic to the group $U \bigl( G(a (A)^{\rm o} ) \bigl)$ of all invertible group-like elements of the finite dual $a (A)^{\rm o}$. For a group $G$, all $G$-gradings on $A$ are explicitly described and classified: the set of isomorphisms classes of all $G$-gradings on $A$ is in bijection with the quotient set $ {\rm Hom}_{\rm BiAlg} \, \bigl( a (A) , \, k[G] \bigl)/\approx$ of all bialgebra maps $a (A) \, \to k[G]$, via th…

Index of Hadamard multiplication by positive matrices II

Given a definite nonnegative matrix $A \in M_n (C)$, we study the minimal index of A: $I(A) = \max \{λ\ge 0 : A\circ B \ge λB$ for all $0\le B\}$, where $A\circ B$ denotes the Hadamard product $(A\circ B)_{ij} = A_{ij} B_{ij}$. For any unitary invariant norm N in $M_n(C)$, we consider the N-index of A: $I(N,A) = \min\{N(A\circ B) : B\ge 0$ and $N(B) = 1 \}$. If A has nonnegative entries, then $I(A) = I(\| \cdot \|_{sp}, A)$ if and only if there exists a vector u with nonnegative entries such that $Au = (1, >..., 1)^T$. We also show that $I(\| \cdot \|_{2}, A)= I(\| \cdot \|_{sp}, {\bar A}\circ…

带对合半群与环中一类新的广义逆

A new class of generalized inverses in semigroups and rings with involution

本文在带对合半群和环中引入 $w$-核逆与对偶 $v$-核逆两类新广义逆，建立其刻画、表达式及与已知广义逆的联系，并给出 $*$-环中基于单位的存在性判据。

Perron相似性与非负逆特征值问题

Perron similarities and the nonnegative inverse eigenvalue problem

本文发展复Perron相似性理论，通过研究I型Karpelevich弧实现矩阵的Perron相似性生成大量可实现谱，验证4×4随机矩阵谱的几何特性，还得到DFT矩阵等相关多胞体的极值性质，推进非负逆特征值问题研究。