四元数矩阵的实表示与四元数矩阵方程

四元数矩阵与四元数矩阵方程在力学和工程问题的理论研究和实际数值计算中都起到重要的作用.该文借助四元数矩阵的实表示方法,研究了一般四元数矩阵方程AXB-CYD=E的解的问题,给出了一种求解四元数矩阵方程的算法技巧.该文还得到了四元数矩阵的Roth's定理.     

奇异四元数矩阵的正态及Wishart分布

本文利用拉直算子(vec)和Kronecker积求得了四元数矩阵的实表示矩阵的一些性质,在此基础上利用实矩阵的奇异正态分布密度函数,求出了四元数矩阵的奇异正态分布的密度函数表达式.由此得到四元数矩阵奇异Wishart分布的密度函数表达式.     

关于四元数矩阵之迹的几个定理

R.Rellman对两个正定实矩阵建立了与Cauchy—Schwarz不等式相类似的结果,引起人们的关注,对Rellman不等式进行深入的研究.但对四元数矩阵之迹的研究至今未见.如所熟知,四元数体的非交换性,已经给四元数代数理论的研究带来了巨大的困难,它也必然影响到四元数矩阵迹的性质.事实上,关于实(或复)矩阵的几个简单性质:     

四元数矩阵方程的复转化及保结构算法

给出四元数矩阵复表示运算定义及其相关性质,并运用复表示运算的保结构特性,讨论了四元数矩阵Moore-Penrose逆计算以及两类四元数矩阵方程AXB=C和AX-XB=C的数值求解方法.数值算例检验了所给算法的可行性.     

八元数矩阵的行列式及其性质

赋范的可除代数只有四种:实数R,复数C,四元数日和八元数O.由于八元数关于乘法非交换且非结合,如何对八元数矩阵定义行列式并使其具有较好的运算性质变得非常困难.最近,李兴民和黎丽根据"八元数自共轭矩阵的行列式应为实数"这一数学与物理上的需求,通过选择几个八元数乘积的次序和结合方式,首次给出了八元数行列式的定义.但是,与实数、复数以及四元数的相应的情形比较,如此定义的行列式,其所具备的运算性质较少.本文给出了一种新的八元数行列式的定义,它们具备了尽可能多的运算性质,同时使得"八元数自共轭矩阵的行列式为实数"不证自明.     

自共轭四元数循环矩阵的特征值问题

循环矩阵是一类应用广泛的特殊矩阵.设A是一个自共轭四元数循环矩阵,运用四元数矩阵的复表示,以及循环矩阵的特定结构形式,得到了矩阵A的特征值的计算公式.反之,对于任意给定的n个实数,证明了一定存在自共轭四元数循环矩阵A,使得A以这n个实数为它的特征值,同时给出了自共轭四元数循环矩阵A的计算方法.推广了复循环矩阵的相关理论结果.     

四元数矩阵凸函数及其不等式

本文推广复矩阵凸函数及其性质到四元数体上矩阵，并且得到关于四元数矩阵凸函数的一些不等式．     

一对四元数矩阵方程的共轭延拓解

定义广义四元数共轭延拓矩阵的概念,利用矩阵分块和四元数矩阵的实表示方法,分别给出四元数矩阵方程AX=C和XB=D存在列共轭延拓解和行共轭延拓解的必要充分条件及解的表达式.     

四元数矩阵方程(AXB,CXD)=(E,F)的最小范数最小二乘Hermitian解

提出了研究四元数矩阵方程(AXB, CXD)=(E, F)的最小范数最小二乘Hermitian解的一个有效方法.首先应用四元数矩阵的实表示矩阵以及实表示矩阵的特殊结构,把四元数矩阵方程转化为相应的实矩阵方程,然后求出四元数矩阵方程(AXB, CXD)=(E, F)的最小二乘Hermitian解集,进而得到其最小范数最小二乘Hermitian解.所得到的结果只涉及实矩阵,相应的算法只涉及实运算,因此非常有效.最后的两个数值例子也说明了这一点.     

一四元数矩阵方程组的最佳逼近解

利用四元数矩阵的广义Frobenius范数和弱圈积,建立一个关于四元数矩阵的实函数并简洁表征其极小值.再用四元数矩阵的奇异值分解和广义Frobenius范数的性质,讨论四元数矩阵方程组[AX,XB]=[C,D]的最小二乘解,得到了解的具体表达式.最后在该方程组的解集合中导出了与给定矩阵的最佳逼近解的表达式.     

Computing the matrix exponential and other matrix functions

In this note we present an algorithm for the computation of matrix functions, especially the matrix exponential. It is a new application of the so-called ‘elimination method’, which was presented for the scalar case in previous papers (e.g. [5]).     

The sign matrix and the separation of matrix eigenvalues

The sign matrices uniquely associated with the matrices (M ? ζ_jI)², where ζ_j are the corners of a rectangle oriented at π/4 to the axes of a Cartesian coordinate system, may be used to compute the number of eigenvalues of the arbitrarily chosen matrix M which lie within the rectangle, and to determine the left and right invariant subspaces of M associated with these eigenvalues. This paper is concerned with the proof of this statement, and with the details of the computation of the required sign matrices.     

A factorization of the symmetric Pascal matrix involving the Fibonacci matrix

In this short note, we give a factorization of the Pascal matrix. This result was apparently missed by Lee et al. [Some combinatorial identities via Fibonacci numbers, Discrete Appl. Math. 130 (2003) 527-534].     

Spectral decomposition of a matrix using the generalized sign matrix

An algorithm for spectral decomposition is presented which does not require knowledge of eigenvalues and eigenvectors. A set of eigenprojectors are defined which covers the entire spectrum of a matrix, and special attention is given to the projection on the zero eigenvalue. Some useful applications are discussed in the paper.     

On the best approximation matrix problem and matrix Fourier series

In this paper the concept of positive definite bilinear matrix moment functional, acting on the space of all the matrix valued continuous functions defined on a bounded interval [a,b] is introduced. The best approximation matrix problem with respect to such a functional is solved in terms of matrix Fourier series. Basic properties of matrix Fourier series such as the Riemann—Lebesgue, matrix property and the bessel—parseval matrix inequality are proved. The concept of total set with respect to a positive definite matrix functional is introduced, and the totallity of an orthonormal sequence of matrix polynomials with respect to the functional is established.     

LDU decomposition of an extension matrix of the Pascal matrix

Let Ω_m,n(α,β,γ) denote a set of all elements of weighted lattice paths with weight (α,β,γ) in the xy-plane from (0,0) to (m,n) such that a vertical step V=(0,1), a horizontal step H=(1,0), and a diagonal step D=(1,1) are endowed with weights α,β, and γ respectively and let ω(Ω_m,n(α,β,γ)) denote the weight of Ω_m,n(α,β,γ) defined by

     

On the best approximation matrix problem for integrable matrix functions

In this paper positive definite matrix functionals defined on a set of square integrable matrix valued functions are introduced and studied. The best approximation problem is solved in terms of matrix Fourier series. Riemann-Lebesgue matrix property and a Bessel-Parseval matrix inequality are given.