关于矩阵幂的秩不变的充要条件的一个证明

设P为一般数域，A=(a_ij)_n×n为P上的矩阵，R(A)为A的秩，利用矩阵的有理标准形给出了A满足R(A²)=R(A)的充分必要条件的一个新的证明.     

通过纯有理运算确定矩阵的Jordan块结构

本文讨论如何通过有限步有理运算求得给定矩阵的Jordan块结构(JBS),因为有理运算可以通过符号计算精确实现.与此对照,迄今为止用数值计算求矩阵的JBS与理论结果相距甚远.证明是构造性的,分两大部分:1)确定矩阵A的不变因子,2)根据A的不变因子确定初等因子结构.为求得A的不变因子,我们提出一种新的Las Vegas算法.它是一种概率型算法,这种算法允许失败,但是当且仅当求得正确答案时才停止运算;     

矩形网格上一类二元有理插值问题

本文首先利用Ｖａｎｄｅｒｍｏｎｄｅ矩阵得到矩形网格上二元多项式插值公式,然后利用该公式建立一类二元有理插值问题的存在性判别准则及有理插值函数的表现公式,并给出数值例子     

矩阵初等变换及其逆阵

本文讨论的矩阵A为数域P上的可逆方阵,对A作初等变换: (i)对调i,j两行(列),这相当于用初等方阵     

关于对广义的正定矩阵进一步研究

通常讨论矩阵的正定性只局限在实对称矩阵范围内(以下我们把全体n阶实对称正定矩阵的集合记为S~+),随着数学本身的发展和其它学科的需要,有不少人开始研究未必对称的较广义的实正定矩阵.李炯生在文[1]中给出了一类较广义的实正定矩阵的定义: 设A是n阶实方阵.如果对于任何非零的n维列向量X都有 X~TAX>0,其中X~T表示X的转置,则把A叫做正定矩阵.全体这类矩阵的集合记为P(I).文[1]证明了A∈P(I)的充分必要条件是A的对称分量是对称正定矩阵(即把A表示为对称矩阵与反对称阵的和的形式,前者称为对称分量,后者称为反对称分量).同时还推得P(I)中矩阵其     

具有k重零特征根的矩阵的性质

本文对具有 k重零特征根的矩阵的一些性质进行了探讨 ,这些性质主要涉及到矩阵的秩、矩阵的有理标准形和约当标准形等方面     

On Compact Graphs

Let G be a finite simple graph with adjacency matrix A, and let P（A） be the convex closure of the set of all permutation matrices commuting with A. G is said to be compact if every doubly stochastic matrix which commutes with A is in P（A）. In this paper, we characterize 3-regular compact graphs and prove that if G is a connected regular compact graph, G - v is also compact, and give a family of almost regular compact connected graphs.     

Lagrange基函数的复矩阵有理插值及连分式插值

1引言 矩阵有理插值问题与系统线性理论中的模型简化问题和部分实现问题有着紧密的联系~[1][2]，在矩阵外推方法中也常常涉及线性或有理矩阵插值问题~[3]。按照文~[1]的阐述。目前已经研究的矩阵有理插值问题包括矩阵幂级数和Newton-Pade逼近。Hade逼近，联立Pade逼近，M-Pade逼近，多点Pade逼近等。显然，上述各种形式的矩阵Pade逼上梁山近是矩     

相似变换矩阵的简单求法

在研究矩阵相似问题时,如果知道矩阵A及相似变换矩阵P,则可求出与A相似的矩阵B=P~(-1)AP 反过来,如果知道A及其相似矩阵B,如何求相似变换矩阵P的问题,一般线性代数教材都很少提及它。即使个别教材中提到这个问题,也只是针对B是A的Jordan标准形的简单情形,应用解非齐次线性方程组AX=XB的方法求出相似变换矩阵P的,因B是特殊情形,所以这种方法不具有普遍意义。     

矩形网格上一类二元有理插值问题

本首先利用Vandermonde矩阵得到矩形网格上二元多项式插值公式．然后利用该公式建立一类二元有理插值问题的存在性判别准则及有理插值函数的表现公式,并给出数值例于。     

Logarithms of iteration matrices, and proof of a conjecture by Shadrin and Zvonkine

A proof of a conjecture by Shadrin and Zvonkine, relating the entries of a matrix arising in the study of Hurwitz numbers to a certain sequence of rational numbers, is given. The main tools used are iteration matrices of formal power series and their (matrix) logarithms.     

Rational Lanczos approximations to the matrix square root and related functions

We consider restricted rational Lanczos approximations to matrix functions representable by some integral forms. A convergence analysis that stresses the effectiveness of the proposed method is developed. Error estimates are derived. Numerical experiments are presented. Copyright © 2008 John Wiley & Sons, Ltd.     

Completely normal matrices

Normal matrices in which all submatrices are normal are said to be completely normal. We characterize this class of matrices, determine the possible inertias of a particular completely normal matrix, and show that real matrices in this class are closed under (general) Schur complementation. We provide explicit formulas for the Moore–Penrose inverse of a completely normal matrix of size at least four. A result on irreducible principally normal matrices is derived as well.     

Spectral methods for orthogonal rational functions

We present an operator theoretic approach to orthogonal rational functions based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure. Two alternatives are discussed, leading to representations which are linear fractional transformations with matrix coefficients acting on infinite Hessenberg or five-diagonal unitary matrices. This approach permits us to recover the orthogonality measure throughout the spectral analysis of an infinite matrix depending uniquely on the poles and the parameters of the recurrence relation for the orthogonal rational functions. Besides, the zeros of the orthogonal and para-orthogonal rational functions are identified as the eigenvalues of matrix linear fractional transformations of finite Hessenberg or five-diagonal matrices. As an application we use operator perturbation theory results to obtain new relations between the support of the orthogonality measure and the location of the poles and parameters of the recurrence relation for the orthogonal rational functions.     

Symmetric Weighing Matrices Constructed using Group Matrices

A weighing matrix of order n and weight m² is a square matrix M of order n with entries from {-1,0,+1} such that MM^T=m²I where I is the identity matrix of order n. If M is a group matrix constructed using a group of order n, M is called a group weighing matrix. Recently, group weighing matrices were studied intensively, especially when the groups are cyclic and abelian. In this paper, we study the abelian group weighing matrices that are symmetric, i.e.M^T=M. Some new examples are found. Also we obtain a few exponent bounds on abelian groups that admit symmetric group weighing matrices. In particular, we prove that there is no symmetric abelian group weighing matrices of order 2p^r and weight p² where p is a prime and p≥ 5.Communicated by: K.T. Arasu     

Production Matrices and Riordan Arrays

We translate the concept of succession rule and the ECO method into matrix notation, introducing the concept of production matrix. This allows us to combine our method with other enumeration techniques using matrices, such as the method of Riordan matrices. Finally we treat the case of rational production matrices, i.e., those leading to rational generating functions. L. Ferrari and S. Rinaldi have been partially supported by MIUR project: Linguaggi formali e automi: metodi, modelli e applicazioni.     

关于正定矩阵一不等式的简单证明

设A=(aij)是一n阶正定实对称矩阵,本文用代数方法证明了|A|≤a11a22…ann,当且仅当A是对角矩阵时等号成立.且证法简单.     

On a number of poisson matrices in Bang-Bang representations for 3 × 3 embeddable matrices

We give a counterexample to the Strong Bang-Bang Conjecture according to which any 3 × 3 embeddable matrix can be expressed as a product of six Poisson matrices. We exhibit a 3 × 3 embeddable matrix which can be expressed as a product of seven but not six Poisson matrices. We show that an embeddable 3 × 3 matrix P with det $\text{P ≥}\text{1}\text{8}$ can be expressed as a product of at most six Poisson matrices and give necessary and sufficient conditions for a 3 × 3 stochastic matrix P with det $\text{P ≥}\text{1}\text{8}$ to be embeddable. For an embeddable 3 × 3 matrix P with det $\text{P <}\text{1}\text{8}$ we give a new bound for the number of Poisson matrices in its Bang-Bang representation.     

Invariant factors, Wiener-Hopf factorization indices, and invariant factors at infinity of rational matrices

The relationship between the finite structure, the infinite structure, and the Wiener-Hopf factorization indices of any rectangular rational matrix is studied.     

层次分析正互反矩阵的Hadamard乘积分析法

定义了标准形 ～ 型 ,并指出任一正互反矩阵可唯一分解为任一种标准形和一个一致性矩阵的Hadamard乘积 .