Generators of matrix algebras in dimension 2 and 3

Let K be an algebraically closed field of characteristic zero and consider a set of 2×2 or 3×3 matrices. Using a theorem of Shemesh, we give conditions for when the matrices in the set generate the full matrix algebra.     

Analyzing priority queues with 3 classes using tree-like processes

In this paper we demonstrate how tree-like processes can be used to analyze a general class of priority queues with three service classes, creating a new methodology to study priority queues. The key result is that the operation of a 3-class priority queue can be mimicked by means of an alternate system that is composed of a single stack and queue. The evolution of this alternate system is reduced to a tree-like Markov process, the solution of which is realized through matrix analytic methods. The main performance measures, i.e., the queue length distributions and loss rates, are obtained from the steady state of the tree-like process through a censoring argument. The strength of our approach is demonstrated via a series of numerical examples. AMS Subject Classifications Primary—60K25; Secondary—60M20, 90B22     

矩阵方程AXC＋BYD＝E的解及其最佳逼近

本文利用矩阵的广义奇异值分解给出了矩阵方程AXC＝BYD＝E有解的充分必要条件及其通解的表达式，同时在矩阵方程的解集合中导出了与给定矩阵的最佳逼近解的表达式。     

两类r-循环矩阵求逆的快速算法

本文利用多项式的最大公因式给出的求r-循环矩阵和对称r-循环矩阵求逆的快速算法。该方法不需要计算三角函数并且具有很少的计算量。     

格矩阵的行列式与伴随矩阵

研究了格矩阵的行列式与伴随矩阵,给出了它们的一些代数性质,同时给出了由一个格矩阵构造一个传递矩阵的方法.     

k-控制阵

给出k-控制阵的定义,讨论k-控制阵与k-圈控制阵的关系,指出k-控制阵的周期是k的一个因子,指数不大于(n-1)k m.     

The Minimum Rank of a 3 × 3 Partial Block Matrix

After recalling the definition and some basic properties of finite hypergroups—a notion introduced in a recent paper by one of the authors—several non-trivial examples of such hypergroups are constructed. The examples typically consist of n n×n matrices, each of which is an appropriate polynomial in a certain tri-diagonal matrix. The crucial result required in the construction is the following: ‘let A be the matrix with ones on the super-and sub-diagonals, and with main diagonal given by a ₁…a _n which are non-negative integers that form either a non-decreasing or a symmetric unimodal sequence; then A_k =P_k (A) is a non-negative matrix, where p_k denotes the characteristic polynomial of the top k× k principal submatrix of A, for k=1,…,n. The matrices A_k as well as the eigenvalues of A, are explicitly described in some special cases, such as (i) a_i =0 for all ior (ii) a_i =0 for i<n and a_n =1. Characters ot finite abelian hypergroups are defined, and that naturally leads to harmonic analysis on such hypergroups.     

A singular value inequality for Heinz means

We prove a matrix inequality for matrix monotone functions, and apply it to prove a singular value inequality for Heinz means recently conjectured by Zhan.     

Fuzzy矩阵正则的充分条件

提出了文献[1]中定理1的一种简单证明方法,并证明了文献[1]中定理2的模糊矩阵A正则的必要条件△≠φ也是充分条件,从而得到以中判定模糊矩阵是否正则的简便方法.     

模糊互补判断矩阵的一种一致性调整方法及其收敛性

首先对一个常用的检验互补判断矩阵一致性标准作了简化,对任一模糊互补矩阵构造相应的模糊一致矩阵,在此基础上给出了一种一致性调整的新方法,并证明了此方法的收敛性。最后用一个算例说明这种调整方法的应用。     

Some Spectral Properties of Metapositive Definite Matrice

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粗信息矩阵的数量特征

利用已给出粗信息矩阵及粗集的有关理论,提出并研究了粗信息矩阵的两个数量特征--粗牲度短阵及粗精度矩阵,给出了一系列性质及定理.     

求非亏损矩阵特征值的一种数值计算方法

借助相似变换将非亏损矩阵转为Hessenberg矩阵,通过获得确定Hessenberg矩阵特征多项式系数的方法,利用特征值与特征多项式系数间的关系,给出求非亏损矩阵特征值的一种数值算法。     

Minimum Permanents on a Face of the Polytope of Doubly Stochastic Matrices II

We determine the minimum permanents and minimizing matrices on the face of z_3+n, for the fully indecomposable (0,1) matrices of order 3+n, which include an identity submatrix of order n .     

Minimum Permanents on a Face of the Polytope of Doubly Stochastic Matrices II

We determine the minimum permanents and minimizing matrices on the face of ω_3+n , for the fully indecomposable (0,1) matrices of order 3+n, which include an identity submatrix of order n .     

矩阵方程X＋AXB＝C与线性流形上的矩阵最佳逼近

该文给出了矩阵方程Ｘ＋ＡＸＢ＝Ｃ存在唯一解的充分必要条件和解的表达式,该公式只是Ａ,Ｂ,Ｃ的多项式,利用该结果,解决了Ａ１ＸＢ１－Ｃ的解的表达式问题．     

The geometry of rank decompositions of matrix multiplication II: 3 × 3 matrices

This is the second in a series of papers on rank decompositions of the matrix multiplication tensor. We present new rank 23 decompositions for the $3\times 3$ matrix multiplication tensor $M_{〈3〉}$. All our decompositions have symmetry groups that include the standard cyclic permutation of factors but otherwise exhibit a range of behavior. One of them has 11 cubes as summands and admits an unexpected symmetry group of order 12.We establish basic information regarding symmetry groups of decompositions and outline two approaches for finding new rank decompositions of $M_{〈\mathbf{n}〉}$ for larger n.     

关于Fuzzy矩阵的Moore-Penrose逆

讨论Fuzzy矩阵的Moore-PenrOSe逆,给出一些Moore-Penrose逆存在的充要条件以及Moore-Penrose逆的划画。     

带概率判断和模糊区间判断的一种排序算法

对于 AHP中一类判断为模糊、不确定性问题 ,用随机变量和模糊区间描述其判断 ,采用 0 .1～ 0 .9标度 ,建立模糊互补判断矩阵 ,利用数学变换得到模糊一致性判断矩阵 ,给出排序向量算法及公式 ,便于实际应用