正交矩阵在空间坐标变换中的作用

借助矩阵乘积分解的思想．研究三阶正交矩阵在空间坐标旋转变换中的作用以及正交矩阵的特征值与转轴、转角之间的关系，获得空间任意向量旋转对应的正交矩阵通式，旋转变换的转轴、转角与变换矩阵的特征属性之间的量化关系，向量经过行列式为-1的正交矩阵迭代变换后特殊的分布规律．向量旋转的矩阵表示将有助于其在工程计算和编程方面的应用．     

谈“关于n阶行列式的一个等式”

贵刊1983年9月登载的“关于n阶行列式的一个等式”一文,提出了这样一个定理:(称它作定理G_1)设有n阶(n≥3)行列式     

弱条件多函数对称式柯西中值定理

本文利用行列式作工具,给出了多函数对称式柯西中值定理,减弱了柯西中值定理的条件.     

Cauchy-Binet定理与Laplace定理的等价证明

本文运用分块矩阵及多元多项式的性质对行列式求值中的Cauchy-Binet 定理与Laplace 定理给出了等价证明.     

Cramer法则的推广

在行列式的理论中,有下列两个重要的而且是众所周知的定理: 定理1、任何行列式都可以按任意一行(列)展开,即若a_(i1),a_(i2),…,a_(in)为n阶行列式D的第i行元素,则     

弱条件多函数含高阶导数的对称式柯西中值定理

利用行列式的性质,给出了多函数对称式含高阶导数的柯西中值定理,减弱了柯西中值定理的条件.     

cauchy-Binet定理与Laplace定理的等价证明

本运用分块矩陈及多元多项式的性质对行列式求值中的Cauchy-Binet定理与Laplace定理给出了等价证明。     

矩阵基础上的行列式理论

首先证明矩阵的伴随矩阵存在性定理，其次利用递推定义方法建立行列式概念，由此可构建矩阵基础上的行列式理论。     

基于特征列方法和Wronskian行列式的曲面定理机器证明

将Chou与Gao的关于微分几何中曲线定理机器证明的方法推广到微分几何曲面定理中. 改进了经典的Wronskian行列式, 它可以用于判断微分域中的有限个元素是否在其常数域上线性相关. 基于Wronskian行列式, 可以用代数语言来描述微分几何曲面理论中的几何表述, 进而用特征列方法来证明这些定理.     

二种矩阵分解定理的证明和分解矩阵构造方法的探讨

本文利用上三角阵T的结果证明了正定矩阵的Cholesky分解定理和非奇异矩阵的QR分解定理,并由T得到分解矩阵L,构造出正交矩阵Q和上三角阵R。     

LEVERRIER-LAGUERRE ALGORITHM FOR THE MATRIX POLYNOMIAL OF DEGREE TWO

1　ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＡｂｅａｎ×ｎｍａｔｒｉｘ ,Ｉｎｄｅｎｏｔｅｓｔｈｅｕｎｉｔｍａｔｒｉｘｏｆｏｒｄｅｒｎ .Ａｎｄｌｅｔ ａ(λ) =ｄｅｔ(λＩｎ-Ａ) =λｎ+ ａ1λｎ- 1+… + ａｎ- 1λ+ ａｎ (1 .1 )ｂｅｔｈｅｃｈａｒａｃｔｅｒｉｓｔｉｃｐｏｌｙｎｏｍｉａｌｏｆＡ ,ｔｈｅａｄｊｏｉｎｔｍａｔｒｉｘｏｆλＩｎ-Ａｂｅ Ｂ(λ) =ａｄｊ(λＩｎ-Ａ) =λｎ- 1Ｉｎ+λｎ- 2 Ｂ1+… +λ Ｂｎ- 2 + Ｂｎ- 1. (1 .2 )Ｔｈｅｎ(λΙｎ-Ａ) - 1= Ｂ(λ) / ａ(λ) . (1 .3)　　Ａｗｅｌｌ ｋｎｏｗＬｅｖｅｒｒｉ…     

ESTIMATES OF THE DOUBLE DETERMINANTS OF QUATERNION MATRICES

An estimate of the upper bound is given for the double determinant of the sum of two arbitrary quaternion matrices, and meanwhile the lower bound on the double determinant is established especially for the sum of two quaternion matrices which form an assortive pair. As applications, some known results are obtained as corollaries and a question in the matrix determinant theory is answered completely.     

广义严格对角占优矩阵的几个判定条件

广义严格对角占优矩阵在计算数学、数学物理、控制论等众多领域有着广泛而重要的应用.但实际判断一个矩阵是否为广义严格对角占优矩阵却是困难的.本文利用α-对角占优矩阵的性质,给出了广义严格对角占优矩阵的几个判定条件,扩大了判别范围.     

一类紧支撑矩阵值正交小波的构造

In this paper,we introduce matrix-valued multiresolution analysis and orthogonal matrix-valued wavelets.We obtain a necessary and sufficient condition on the existence of orthogonal matrix-valued wavelets by means of paraunitary vector filter bank theory.A method for constructing a class of compactly supported orthogonal matrix-valued wavelets is proposed by using multiresolution analysis method and matrix theory.     

偶特征正交空间中子空间包含关系的条件及矩阵表示

本文研究了特征为2的有限域上正交空间中子空间的包含关系和子空间的矩阵表示,利用了偶特征正交几何的理论,得到了偶特征正交空间中子空间的包含条件和矩阵表示.     

Matrix Valued Orthogonal Polynomials Arising from the Complex Projective Space

The main purpose of this paper is to display new families of matrix valued orthogonal polynomials satisfying second-order differential equations, obtained from the representation theory of U(n). Given an arbitrary positive definite weight matrix W(t) one can consider the corresponding matrix valued orthogonal polynomials. These polynomials will be eigenfunctions of some symmetric second-order differential operator D only for very special choices of W(t). Starting from the theory of spherical functions associated to the pair (SU(n+1), U(n)) we obtain new families of such pairs {W,D}. These depend on enough integer parameters to obtain an immediate extension beyond these cases.     

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.     

Stiefel–Whitney invariants for bilinear forms

We examine potential extensions of the Stiefel–Whitney invariants from quadratic forms to bilinear forms which are not necessarily symmetric. We show that as long as the symbolic nature of the invariants is maintained, some natural extensions carry only low dimensional information. In particular, the generic invariant on upper triangular matrices is equivalent to the dimension and determinant. Along the process, we show that every non-alternating matrix is congruent to an upper triangular matrix, and prove a version of Witt?s Chain Lemma for upper-triangular bases. (The classical lemma holds for orthogonal bases.)     

A Semi-Conjugate Matrix Boundary Value Problem for General Orthogonal Polynomials on an Arbitrary Smooth Jordan Curve

In this article, the author characterizes orthogonal polynomials on an arbitrary smooth Jordan curve by a semi-conjugate matrix boundary value problem, which is different from the Riemann-Hilbert problems that appear in the theory of Riemann -Hilbert approach to asymptotic analysis for orthogonal polynomials on a real interval introduced by Fokas, Its, and Kitaev and on the unit circle introduced by Baik, Deift, and Johansson. The author hopes that their characterization may be applied to asymptotic analysis for general orthogonal polynomials by combining with a new extension of steepest descent method which we are looking for.     

关于对称矩阵随机变换方差的证明

应用正交变换将对称矩阵对角化,基于随机向量正交变换后独立性的不变性及矩阵迹相关性质,给出一个关于对称矩阵经随机变换后方差的证明,并将该结论推广到更一般情形。