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.     

统一化tame定理

著名的 tame 定理告诉我们, 对于任意的tame bocs和 正整数 $n$, 存在有限多个极小 bocs, 使得原 bocs的任意维数不超过 $n$ 的表示同构于其中某个极小 bocs的一个表示在一定的约化函子之下的像. 本文将用矩阵问题的语言给出 tame 定理的叙述, 并对正整数 $n$ 构造一个统一的极小矩阵问题, 使得原矩阵问题的任意维数不超 过 $n$ 的不可分解表示同构于该极小矩阵问题的一个表示在约化函子之下的像. 同时给出这个不可分解表示的典范形.     

四元数的复数形式及其在6R机器人反解中的应用

把四元数应用于平面旋转的特殊情况,并把旋转角度的正弦、余弦改写为复指数形式,导出平面四元数的两个复数形式的基.利用这两个基可以把表示三维变换的四元数和对偶四元数改造为复数形式,得到复数形式四元数的4个基和复数形式对偶四元数的8个基,以及它们之间乘法运算的运算法则.通过把它们应用到空间$6R$机器人的位移反解问题,证明了Dixon结式展开后的次数为16次,而不是形式上的24次,并且得到单变量的16次方程.     

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.     

Lorentz变换的四元数表示——纪念李国平院士吴新谋教授诞辰100周年

为一般Lorentz变换给出了一种新的形式简单的四元数表示. 其特点是所用四元数的分量要么是实数, 要么是纯虚数. 与以往的向量-张量表示和八元数表示(双四元数)相比, 有其明显的优点.     

抛物型交换四元数矩阵的性质及其逆矩阵求法

以抛物型交换四元数及其矩阵的概念为基础,首先,利用矩阵的计算理论得到了抛物型交换四元数及其实表示的系列性质.其次,推导了抛物型交换四元数矩阵的性质,通过引入矩阵的实表示形式,得到求抛物型交换四元数矩阵逆矩阵的新方法,为进一步研究抛物型交换四元数矩阵的其余问题提供了理论支撑.最后,通过数值例子验证了结论的有效性和正确性.     

Lorentz变换的四元数表示

为一般Lorentz变换给出了一种新的形式简单的四元数表示.其特点是所用四元数的分量要么是实数,要么是纯虚数.与以往的向量-张量表示和八元数表示(双四元数)相比,有其明显的优点.     

四元数矩阵的酉相合与酉相似

本文讨论了四元数矩阵的酉相合,四元数矩阵的酉相合是复矩阵的复酉相合的自然推广,并且它有许多好的性质.由于四元数矩阵的酉相合与酉相似有着密切联系,本文还讨论了四元数矩阵的酉相似的一些性质.     

四元数矩阵方程AXB+CXD=E的广义延拓解

本文在四元数体上讨论矩阵方程AXB+CXD=E的广义行（列）共轭延拓解问题.利用四元数矩阵的复与实分解,以及广义共轭延拓矩阵的结构特点,借助矩阵Kronecker积,把约束四元数矩阵方程转化为实数域上无约束方程,从而得到该方程具有广义行（列）共轭延拓解的充要条件及其通解表达式.最后通过数值算例说明所给算法的可行性.     

广义四元数的一次代数方程

本文运用广义四元数代数的矩阵表示讨论了两类广义四元数的一次代数方程的解问题，并得到了这两类代数方程有唯一解、无穷多解，无解的判别条件。     

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.     

Fundamental Representations and Algebraic Properties of Biquaternions or Complexified Quaternions

The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates, semi-norms, polar forms, and inner products. The notation is consistent throughout, even between representations, providing a clear account of the many ways in which the component parts of a biquaternion may be manipulated algebraically.     

On the matrix algebra of elliptic biquaternions

In this study, we introduce the concept of elliptic biquaternion matrices. Firstly, we obtain elliptic matrix representations of elliptic biquaternion matrices and establish a universal similarity factorization equality for elliptic biquaternion matrices. Afterwards, with the aid of these representations and this equality, we obtain various results on some basic topics such as generalized inverses, eigenvalues and eigenvectors, determinants, and similarity of elliptic biquaternion matrices. These valuable results may be useful for developing a perfect theory on matrix analysis over elliptic biquaternion algebra in the future.     

Produit de bivecteurs et quaternions complexes

First we calculate the product of two bivectors in vectorial spaceR(p, q) (p andq are integers such thatp+q=n). Second we prove that this product is a quaternion forR(3, 0) and we generalize to finite number of bivectors. Third we prove that this product is a biquaternion forR(1, 3) and we genaralize in the same way. Fourth we prove that some complex quaternions can be connected with real Clifford algebra by choosing correctly the usual imaginary.      

幂零--中心化方法

An nxn complex sign pattern(ray pattern) S is said to be spectrally arbitrary if for every monic nth degree polynomial f(λ) with coefficients from C,there is a complex matrix in the complex sign pattern class(ray pattern class) of S such that its characteristic polynomial is f(λ).We derive the Nilpotent-Centralizer methods for spectrally arbitrary complex sign patterns and ray patterns,respectively.We find that the Nilpotent-Centralizer methods for three kinds of patterns(sign pattern,complex sign pattern,ray pattern) are the same in form.     

Triality,biquaternion and vector representation of the Dirac equation

The triality properties of Dirac spinors are studied, including a construction of the algebra of (complexified) biquaternion. A bilinear law of composition for biquaternion is defined by means of Levi-civita symbol and Lorentzian metric only. It is proved that there exists a vector-representation of Dirac spinors. The massive Dirac equation in the vector-representation is actually self-dual. A chiral transformations for spinors is equivalent to U(1) transformations for corresponding complex vectors. The Dirac’s idea of non-integrable phases is used to study the behavior of massive term.     

Tchebyshev Posets

We construct for each $n$ an Eulerian partially ordered set $T_n$ of rank $n+1$ whose $ce$-index provides a non-commutative generalization of the $n$th Tchebyshev polynomial. We show that the order complex of each $T_n$ is shellable, homeomorphic to a sphere, and that its face numbers minimize the expression $\max_{|x|\leq 1} |\sum_{j=0}^n (f_{j-1}/f_{n-1})\cdot 2^{-j}\cdot (x-1)^j|$ among the $f$-vectors of all $(n-1)$-dimensional simplicial complexes. The duals of the posets constructed have a recursive structure similar to face lattices of simplices or cubes, offering the study of a new special class of Eulerian partially ordered sets to test the validity of Stanleys conjecture on the non-negativity of the $cd$-index of all Gorenstein$^*$ posets.     

Descent properties of biquaternion algebras with involution

We use the pfaffian to study some descent properties of biquaternion algebras with involution of the first kind in arbitrary characteristic.     

Biquaternion (Complexified Quaternion) Roots of −1

The roots of −1 in the set of biquaternions (quaternions with complex components, or complex numbers with quaternion real and imaginary parts) are derived. There are trivial solutions (the complex operator, and any unit pure real quaternion), and non-trivial solutions consisting of complex numbers with perpendicular pure quaternion real and imaginary parts. The moduli of the two perpendicular pure quaternions are expressible by a single parameter by using a hyperbolic trigonometric identity.     

Conformal Mappings, Hyperanalyticity and Field Dynamics

Generalization of complex analysis to the case of noncommutative algebras of a quaternion-like type is presented. There exists a correspondence between quaternion-differentiable functions and conformal mappings in Euclidean 4-space. For the algebra of biquaternions differentiability conditions are nonlinear and Lorentz-invariant. Starting from these, a version of algebraic field theory, algebrodynamics is suggested. Solutions of basic equations are obtained, and relations to the Maxwell theory disputed.