四元数矩阵Jordan标准的简单证明和算法

引入了友向量的概念 ,给出了四元数矩阵的秩的一种定义方法和四元数矩阵Jordan标准形的一种简单证明和算法 .     

四元数体上若干线性代数问题的显式

通过建立四元数乘积的一个弱可交换律,分别给出四元数体上的线性方程组的解和克莱姆解式、向量的相关性、矩阵的逆与秩以及线性变换的特征根与特征向量等存在性的充要条件,从而得到这些问题的一种有效计算方法．     

一四元数矩阵方程组的实部半正定解

设Ａ是一个ｎ阶实四元数方阵，若对任意的非零ｎ元列向量ｘ，有ｘＡｘ的实部非负，则称Ａ是一个实部半正定阵，本文给出了实四元数矩阵方程组。     

四元数矩阵的广义Schmidt分解与广酉空间中向量组的广义标准正交化

推广了四元数矩阵的Schmidt分解及广酉空间中向量组的标准正交化问题，给出了实四元数矩阵分解为广酉矩阵与生对角元全正的上三角阵乘积的实用方法．     

分裂四元数矩阵的奇异值分解及应用

利用i-共轭重新定义了分裂四元数矩阵的共轭转置,在此基础上借助复表示和友向量研究了分裂四元数矩阵的奇异值分解,并利用所得结果解决了分裂四元数矩阵的极分解和分裂四元数矩阵方程AXB-CYD=E.     

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．     

四元数体上的矩阵及其优化理论

本文引入了四元数体 Q 上的广义双随机矩阵,给出了它与优化的关系.由此,我们得出了四元数矩阵奇异值的一些重要不等式,特别是得出了四元数矩阵的和与积的奇异值不等式.我们还讨论了四元数自共轭矩阵的和与积的特征值等.推广了复数域上矩阵的许多著名结果.     

四元数体上矩阵的广义对角化

引入了复四元数环和四元数体上矩阵可 对角化的概念,研究了复四元数环上矩阵的性质,给出了四元数体上矩阵可 对角化的充分必要条件和求矩阵 对角化的方法。     

四元数矩阵实表示的基本性质及应用

在四元数实矩阵表示的基础上,给出了四元数矩阵的相同表示,利用友向量的概念,给出了这种实表示的性质,并进一步研究了四元数力学中的系列数值计算问题.     

基于广义的Sylvester实四元数矩阵方程（英文）

基于广义Sylvester实圆元数矩阵方程组的解■当A_i,B_i和C_i(i=1,2,3)是被复数矩阵给定的,X,Y,Z和W是可变矩阵.计算耦合广义S_ylvester实四元数矩阵方程组的通解W的秩的极值.     

Pseudoquaternion Geometry

We describe complex holomorphic transformations of a quaternion vector space taking left quaternion lines to left quaternion lines and real linear transformations of the quaternion plane simultaneously preserving the sets of left and right quaternion lines.     

四元数体上一类矩阵方程解的数值方法

建立了求解四元数体上严格对角占优矩阵方程AX=B的QJ和QSOR迭代方法,并利用四元数矩阵的右特征值最大模刻画出迭代的收敛性,给出参数的取值范围;最后运用四元数矩阵的复表示运算保结构的特性,把这两种迭代等价地转化到复数域上,从而实现了该系统的数值求解.     

On the eigenvalues of quaternion matrices

This article is a continuation of the article [F. Zhang, Ger?gorin type theorems for quaternionic matrices, Linear Algebra Appl. 424 (2007), pp. 139–153] on the study of the eigenvalues of quaternion matrices. Profound differences in the eigenvalue problems for complex and quaternion matrices are discussed. We show that Brauer's theorem for the inclusion of the eigenvalues of complex matrices cannot be extended to the right eigenvalues of quaternion matrices. We also provide necessary and sufficient conditions for a complex square matrix to have infinitely many left eigenvalues, and analyse the roots of the characteristic polynomials for 2?×?2 matrices. We establish a characterisation for the set of left eigenvalues to intersect or be part of the boundary of the quaternion balls of Ger?gorin.     

Solution to fuzzy system of linear equations with crisp coefficients

This paper presents a new and simple method to solve fuzzy real system of linear equations by solving two n × n crisp systems of linear equations. In an original system, the coefficient matrix is considered as real crisp, whereas an unknown variable vector and right hand side vector are considered as fuzzy. The general system is initially solved by adding and subtracting the left and right bounds of the vectors respectively. Then obtained solutions are used to get a final solution of the original system. The proposed method is used to solve five example problems. The results obtained are also compared with the known solutions and found to be in good agreement with them.     

Some remarks on one-sided regularity

A ring is said to be right (resp., left) regular-duo if every right (resp., left) regular element is regular. The structure of one-sided regular elements is studied in various kinds of rings, especially, upper triangular matrix rings over one-sided Ore domains. We study the structure of (one-sided) regular-duo rings, and the relations between one-sided regular-duo rings and related ring theoretic properties.     

Pseudocomplex and Pseudo(bi)quaternion Mappings

A problem posed by V. I. Arnold is solved by describing all homeomorphisms of an affine quaternion space that take any left or right affine quaternion line to a left or right affine quaternion line (a left line can be taken to a right one, and vice versa).     

Involutions of Complexified Quaternions and Split Quaternions

An involution or anti-involution is a self-inverse linear mapping. Involutions and anti-involutions of real quaternions were studied by Ell and Sangwine [15]. In this paper we present involutions and antiinvolutions of biquaternions (complexified quaternions) and split quaternions. In addition, while only quaternion conjugate can be defined for a real quaternion and split quaternion, also complex conjugate can be defined for a biquaternion. Therefore, complex conjugate of a biquaternion is used in some transformations beside quaternion conjugate in order to check whether involution or anti-involution axioms are being satisfied or not by these transformations. Finally, geometric interpretations of real quaternion, biquaternion and split quaternion involutions and anti-involutions are given.     

On additively completely regular seminearrings

In an attempt to investigate the situation arising out of replacing additive regularity by additive complete regularity in our previous study on additively regular seminearrings, we introduce the notions of left (right) completely regular seminearrings and characterize left (right) completely regular seminearrings as bi-semilattices of left (resp., right) completely simple seminearrings. We also define left (right) Clifford seminearrings and show that they are precisely bi-semilattices of near-rings (resp., zero-symmetric near-rings).