广义酉矩阵与广义Hermite矩阵

给出了广义酉矩阵与广义(斜)Hermite矩阵的概念，研究了它们的性质及其与酉阵、共轭辛阵、Hermite阵、Hamilton及广义逆矩阵之间的联系；取得了许多新的结果；推广了酉矩阵、Hermite阵与斜Hermite阵间的相应结果，特别将正交阵的广义Cayley分解推广到了广义酉矩阵上；将各类酉矩阵、Hermite矩阵及广义逆矩阵统一了起来．     

可逆矩阵的高次伴随矩阵

用数学归纳法推出了可逆矩阵的高次伴随矩阵的公式,并结合可逆矩阵的基本公式得出了可逆矩阵的高次伴随矩阵的行列式和逆矩阵,给出了可逆矩阵的高次伴随矩阵的特征值和特征向量的表示公式,最后讨论了若干个可逆矩阵的乘积的高次伴随矩阵.     

Matrix Christoffel Functions

We introduce and study matrix Christoffel functions for a matrix weight W. We find an explicit expression of the matrix Christoffel functions in terms of any sequence of orthonormal matrix polynomials with respect to W. An extremal property related to the matrix moment problem defined by W is established for the matrix Christoffel functions. We finally find the relative asymptotic behavior of the matrix Christoffel functions associated to matrix weights in the matrix Nevai class.     

几类特殊矩阵的性质研究及应用

阐述了判断矩阵、度量矩阵及偏离矩阵的特征值相同性以及它们的特征向量的关系.用度量矩阵和判断矩阵的偏离矩阵作比较,这能够清楚地表明哪些元素对判断矩阵的一致性影响较大.     

ON HERMITE MATRIX POLYNOMIALS AND HERMITE MATRIX FUNCTIONS

In this paper properties of Hermite matrix polynomials and Hermite matrix functions are studied. The concept ot total set with respect to a matrix functional is introduced and the total property of the Hermite matrix polynomials is proved. Asymptotic behaviour of Hermite matrix polynomials is studied and the relationship of Hermite matrix functions with certain matrix differential equations is developed. A new expression of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.     

基于部分基变量的LP问题矩阵算法

基于部分基变量提出了LP问题的矩阵算法. 该算法以最优基矩阵的一个充分必要条件为基础,首先将一个初始矩阵转化为右端项和检验数均满足要求的矩阵,再转为检验数满足要求的基矩阵,最后转化为最优基矩阵.该算法具有使用范围广、计算规模小、计算过程简化、计算机易于实现的优势.矩阵算法的核心运算是求逆矩阵的运算,提出了矩阵算法的求逆问题,讨论并给出了求逆快速算法,该算法充分利用了矩阵算法迭代过程中提供的原来的逆矩阵的信息经过简单的变换得到新的逆矩阵,该算法比直接求逆法计算效率更高.     

Inverting linear combinations of identity and generalized Catalan matrices

We introduce the notion of the generalized Catalan matrix as a kind of lower triangular Toeplitz matrix whose nonzero elements involve the generalized Catalan numbers. Inverse of the linear combination of the Pascal matrix with the identity matrix is computed in Aggarwala and Lamoureux (2002) [1]. In this paper, continuing this idea, we invert various linear combinations of the generalized Catalan matrix with the identity matrix. A simple and efficient approach to invert the Pascal matrix plus one in terms of the Hadamard product of the Pascal matrix and appropriate lower triangular Toeplitz matrices is considered in Yang and Liu (2006) [14]. We derive representations for inverses of linear combinations of the generalized Catalan matrix and the identity matrix, in terms of the Hadamard product which includes the Generalized Catalan matrix and appropriate lower triangular Toeplitz matrix.     

关于k-广义酉矩阵

本文研究了k-广义酉矩阵的性质及其与酉矩阵、辛矩阵、Householder矩阵之间的联系,取得了许多新的结果,推广了酉矩阵及Householder矩阵的相应结果,特别将正交矩阵的广义Cayley分解推广到了广义酉矩阵上;并将各类酉矩阵及辛矩阵统一了起来.     

关于正定矩阵的子式阵的正定性

研究正定矩阵的子矩阵,利用合同标准形分别给出了复正定矩阵的子式阵为复正定矩阵和实正定矩阵的子式阵为实正定矩阵的充分必要条件,其结果简单而实用.     

Necessary and sufficient conditions for a matrix to be causative in a nonstationary Markov chain

Given a sequence of transition matrices for a nonstationary Markov chain, a matrix whose product on the right of a transition matrix yields the next transition matrix is called a causative matrix. A causative matrix is strongly causative if successive products continue to yield stochastic matrices. This paper presents necessary and sufficient conditions for a matrix to be causative and strongly causative with respect to an invertible transition matrix, by considering the causative matrix as a linear transformation on the rows of the transition matrix.     

Partitioning of invariant imbedding systems

The choice of partitioning the system matrix for a system of N linear ordinary differential equations may determine the ease or difficulty of obtaining a solution by the invariant imbedding method of Scott. This paper shows how the configuration and partitioning of the system matrix is reflected in the fundamental matrix. The partitioning of the fundamental matrix is the key to the ease or difficulty of obtaining a solution. If the fundamental matrix is known for a given system matrix configuration and partitioning, then the fundamental matrix associated with a new system matrix configuration may be derived by the same row and column interchanges that transformed the old system matrix into the new system matrix. The fundamental matrix for the new system matrix does not have to be recalculated anew from the Kronecker delta initial conditions.     

ORTHOGONAL MATRIX POLYNOMIALS WITH RESPECT TO A CONJUGATE BILINEAR MATRIX MOMENT FUNCTIONAL: BASIC THEORY

In this paper basic results for a theory of orthogonal matrix polynomials with respect to a conjugate bilinear matrix moment functional are proposed. Properties of orthogonal matrix polynomial sequences including a three term matrix relationship are given. Positive definite conjugate bilinear matrix moment functionals are introduced and a characterization of positive definiteness in terms of a block Haenkel moment matrix is established. For each positive definite conjugate bilinear matrix moment functional an associated matrix inner product is defined.     

次正规矩阵、次酉矩阵、次厄米特矩阵及反次厄米特矩阵

主要研究了下列几方面问题:(i)次酉矩阵、次厄米特矩阵及反次厄米特矩阵的特征值与次特征值;(ii)次正规矩阵、次酉矩阵、次厄米特矩阵及反次厄米特矩阵分别与正规矩阵、酉矩阵、厄米特矩阵及反厄米特矩阵之间的关系;(iii)次正规矩阵、次酉矩阵、次厄米特矩阵及反次厄米特矩阵之间的联系.     

确定概率条件下风险型多属性消错决策方法

以确定概率条件下风险型多属性决策问题为研究对象.根据消错理论提出了错误值、极限损失值等概念,以效益矩阵为基础建立起正负理想矩阵和错误值矩阵,以正负理想矩阵为基础构建极限损失矩阵,以错误值矩阵、属性权重和极限损失矩阵为基础构建综合错误损失矩阵.接着根据期望理论,利用综合错误损失矩阵求取期望错误损失向量,并以此作为策略选择的根据.最后通过实例证明了研究的有效性和可行性.     

相似矩阵与等价矩阵的收敛性及其传递闭包

利用模糊矩阵的有向伴随图,探讨了模糊相似矩阵与模糊等价矩阵的收敛性,证明了模糊相似矩阵与模糊等价矩阵的收敛指数.然后通过计算模糊相似矩阵的传递闭包,给出了改造模糊相似矩阵使之成为模糊等价矩阵的一条简单途径.最后通过一个关于模糊聚类分析的算例,展现了基于模糊相似矩阵的传递闭包法在农业蝗灾防控中的应用.     

完全单的矩阵半群

引入矩阵型Rees矩阵半群的概念,证明完全单的矩阵半群等价于矩阵型Rees矩阵半群,进而给出矩阵拓扑半群的极小理想的刻画以及完全正则矩阵半群特别是一些重要类别的群带的刻画．     

一类非线性矩阵方程对称解的双迭代算法

利用逆矩阵的Neumann级数形式,将在Schur插值问题中遇到的含未知矩阵二次项之逆的非线性矩阵方程转化为高次多项式矩阵方程,然后采用牛顿算法求高次多项式矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立求非线性矩阵方程的对称解的双迭代算法.双迭代算法仅要求非线性矩阵方程有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定.数值算例表明,双迭代算法是有效的.     

广义循环矩阵的一个性质

可表为非奇异对角矩阵和循环矩阵乘积的矩阵,我们称其为广义循环矩阵.本文给出了单位矩阵与广义循环矩阵的和矩阵的非奇异的充要条件,得到了这样和矩阵的相对增益阵列的显示表达式.     

广义严格对角占优阵的判定程序

1 引言和符号 在本文中,均采用下列符号而不再重申.恒用N表示前n个自然数的集合;而用Mn(C)和Mn(R)分别表示所有n阶复矩阵和所有n阶实矩阵的集合. Z_N={A|A=(a_(ij))_(n×n)∈Mn(R),a_(ij)≤0,i,j∈N,i≠j},I恒表示单位矩阵. 如果A∈Mn(R)且A的所有元素都为非负实数,则称A为非负方阵,并记为A≥0;若A的所有元素都为正数,则称A为正矩阵,并记为A>0. 对A=(a_(ij))(n×n)∈Mn(C),令A_i(A)=sum from j=1 j≠i to n (|a_(ij)|(i=1、2…… n)) ;若把A的非零元用1代替 而得到—个n阶(0,1)矩阵。称为A的导出矩阵。记为;而把A的比较矩阵记为 u(A)=(b_(ij))_(n×n))其中b_(ij)=|a_(ij)|,b_(ij)=-|a_(ij)|(i,j∈N i≠j)     

图论中矩阵的可实现性

对图的关联矩阵,邻接矩阵,基本割集矩阵,基本圈矩阵的可实现性分别进行了论证,并将邻接矩阵的可实现性推广到一般形式.得到了同一个基本割集矩阵的奥凯达图形是不唯一的;以及这些奥凯达图形所对应的图是互相同构的结果;并且指出了基本圈矩阵的可实现性可以依靠基本割集矩阵的可实现性来解决.