关于三幂等矩阵的秩特征的研究

本文对已有的关于三幂等矩阵秩的等式作了进一步研究，指出其中有些可以作为判定三幂等矩阵的充要条件，即三幂等矩阵的秩特征等式，本文还证明了有无穷多种三幂等矩阵的秩特征等式形式．     

高等代数中一道习题的推广

利用分块矩阵及其初等变换将一类矩阵秩的等式推广.     

两个矩阵秩等式的推广

给出文[1]建立的两个矩阵秩等式的统一推广.     

关于矩阵秩（不）等式的分块矩阵构造证明

利用构造分块矩阵并通过广义初等变换的方法，证明矩阵秩的（不）等式．     

矩阵秩分解定理的一个证明及其应用

矩阵的秩分解定理是矩阵论中的一个基础性定理.本文给出了秩分解定理的一个证明,描述了其中可逆矩阵P,Q的构造,讨论了秩分解定理及其P,Q的构造在解线性方程组中的应用,以及在判别Sylvester不等式等号成立中的应用.     

关于幂等矩阵秩的一个等式

目的:将复数域上幂等矩阵秩的一个等式推广到除环上.方法:采用广义逆矩阵的理论.结果:得到了除环上的类似等式.结论:复幂等矩阵的某些等式是可以推广到除环上的,并且可以进行必要的改进.     

线性方程组解空间的进一步性质及应用

讨论齐次线性方程组解空间的进一步性质,以及在矩阵秩等式证明中的应用.     

浅淡公式rkAB＝rkB—dim（R（B）∩N（A）的应用

本文的目的是给出有关矩阵乘积的秩的一个等式.然后据此研究一系列秩数问题.定理若矩阵A与B可乘,则rkAB=rkB—(dimR(B)∩N(A)) (1) =rkA—dim(R(A′)∩N(B′)) (1′)其中R(B)是B的值域,N(A)是A的零空间;rkA记A的秩,dimR(B)记只(B)的维数.     

矩阵秩的等式的几个证法

矩阵的秩是线性代数的重要概念,本文给出了有关矩阵秩的等式的七种证法.     

M-矩阵的性质与Hadamard-Fisher不等式的注记

主要研究了两个M-矩阵的比较性质与不等式,给出了M-矩阵与逆M-矩阵Hadamard-Fisher不等式等式成立的矩阵结构.     

The Moore-Penrose inverse for sums of matrices under rank additivity conditions

Some results on the Moore-Penrose inverse for sums of matrices under rank additivity conditions are revisited and some new consequences are presented. Their extensions to the weighted Moore-Penrose inverse of sums of matrices under rank additivity conditions are also considered.     

Continuity and separation in symmetric topologies

The well-known Frobenius rank inequality established by Frobenius in 1911 states that the rank of the product ABC of three matrices satisfies the inequality rank(ABC) U rank(AB) + rank(BC)- rank(B) A new necessary and sufficient condition for equality to hold is presented and then some interesting consequences and applications are discussed.     

Abel范畴中态射乘积诱导的短正合列及其应用

本文给出了Ａｂｅｌ范畴中态射乘积αβγ诱导的短正合列的立体交换图。其中共有７个３×３平面，３０条线。每一条线（两端的零对象省略）都是一个短正合列．这些短正合列用到ｐ－除环上的矩阵，可以导出一系列矩阵秩的恒等式，它不仅推广了域上矩阵秩的结果，还能得到一些新的关系式．     

On Rank Invariance of Schwarz-Pick-Potapov Block Matrices of Matricial Schur Functions

We derive statements on rank invariance of Schwarz-Pick-Potapov block matrices of matrix-valued Schur functions. The rank of these block matrices coincides with the rank of some block matrices built from the corresponding section matrices of Taylor coefficients. These results are applied to the discussion of a matrix version of the classical Schur-Nevanlinna algorithm.     

On the minimum semidefinite rank of a simple graph

The minimum semidefinite rank (msr) of a graph is defined to be the minimum rank among all positive semidefinite matrices whose zero/nonzero pattern corresponds to that graph. We recall some known facts and present new results, including results concerning the effects of vertex or edge removal from a graph on msr.     

Fast algorithms for hierarchically semiseparable matrices

Semiseparable matrices and many other rank‐structured matrices have been widely used in developing new fast matrix algorithms. In this paper, we generalize the hierarchically semiseparable (HSS) matrix representations and propose some fast algorithms for HSS matrices. We represent HSS matrices in terms of general binary HSS trees and use simplified postordering notation for HSS forms. Fast HSS algorithms including new HSS structure generation and HSS form Cholesky factorization are developed. Moreover, we provide a new linear complexity explicit ULV factorization algorithm for symmetric positive definite HSS matrices with a low‐rank property. The corresponding factors can be used to solve the HSS systems also in linear complexity. Numerical examples demonstrate the efficiency of the algorithms. All these algorithms have nice data locality. They are useful in developing fast‐structured numerical methods for large discretized PDEs (such as elliptic equations), integral equations, eigenvalue problems, etc. Some applications are shown. Copyright © 2009 John Wiley & Sons, Ltd.     

线性映射方法在矩阵理论和运算中的应用

对矩阵的一些运算关系从映射角度考虑,得到概念上的新理解和运算的新技巧.特别是,给出了Frobenius不等式,Sylvester不等式等著名结果的极其简洁的证明,据此探讨了线性代数中有关问题和实例,包括列满秩矩阵的特点等.     

矩阵值Caratheodory函数广义块Pick矩阵的秩不变性

Lasarow^[1]推导出矩阵值Carath\'{e}odory函数的第一、第二型广义块Pick矩阵及其变型的秩不变性. 这些矩阵由同一个Carath\'{e}odory函数的值与它的直到某阶的导数值确定. 利用文献[2]中提出的块Toeplitz向量方法, 该文断言,这些块矩阵的秩分别相关并重合于具有秩不变性的块Toeplitz矩阵的秩, 从而改进了这两类广义块Pick矩阵的秩不变性结论的证明.     

Estimations of parametric functions under a system of linear regression equations with correlated errors

Estimations of parametric functions under a system of linear regression equations with correlated errors across equations involve many complicated operations of matrices and their generalized inverses. In the past several years, a useful tool -- the matrix rank method was utilized to simplify various complicated operations of matrices and their generalized inverses. In this paper, we use the matrix rank method to derive a variety of new algebraic and statistical properties for the best linear unbiased estimators （BLUEs） of parametric functions under the system. In particular, we give the necessary and sufficient conditions for some equalities, additive and block decompositions of BLUEs of parametric functions under the system to hold.     

非交换主理想整环上分块矩阵的秩

本文从非交换主理想整环Ｒ上矩阵Ａ的秩与它在Ｒ所嵌入的商除环Ｋ上的秩间的关系着手，证得了Ｒ上分块矩阵秩的一些结果，因此也解决了［１］中关于ｐ￣一除环上矩阵秩的一个猜想．