This paper aims to present, in a unified manner, the algebraic techniques of eigen-problem which are valid on both the quaternions and split quaternions. This paper studies eigenvalues and eigenvectors of the v-quater...This paper aims to present, in a unified manner, the algebraic techniques of eigen-problem which are valid on both the quaternions and split quaternions. This paper studies eigenvalues and eigenvectors of the v-quaternion matrices by means of the complex representation of the v-quaternion matrices, and derives an algebraic technique to find the eigenvalues and eigenvectors of v-quaternion matrices. This paper also gives a unification of algebraic techniques for eigenvalues and eigenvectors in quaternionic and split quaternionic mechanics.展开更多
This paper aims to present, in a unified manner, algebraic techniques for linear equations which are valid on both the algebras of quaternions and split quaternions. This paper, introduces a concept of v-quaternion, s...This paper aims to present, in a unified manner, algebraic techniques for linear equations which are valid on both the algebras of quaternions and split quaternions. This paper, introduces a concept of v-quaternion, studies the problem of v-quaternionic linear equations by means of a complex representation and a real representation of v-quaternion matrices, and gives two algebraic methods for solving v-quaternionic linear equations. This paper also gives a unification of algebraic techniques for quaternionic and split quaternionic linear equations in quaternionic and split quaternionic mechanics.展开更多
In this paper, from the spacetime algebra associated with the Minkowski space ℝ3,1by means of a change of signature, we describe a quaternionic representation of the split-tetraquaternion algebra which incorporates th...In this paper, from the spacetime algebra associated with the Minkowski space ℝ3,1by means of a change of signature, we describe a quaternionic representation of the split-tetraquaternion algebra which incorporates the Pauli algebra, the split-biquaternion algebra and the split-quaternion algebra, we relate these algebras to Clifford algebras and we show the emergence of the stabilized Poincaré-Heisenberg algebra from the split-tetraquaternion algebra. We list without going into details some of their applications in Physics and in Born geometry.展开更多
随着四元数的广泛应用,大型四元数结构矩阵方程的求解成为科学计算的重要课题。针对四元数亚正定系统AX=B,在新自共轭正定和斜自共轭分裂迭代(new positive definite and skew-self-conjugate splitting,NPSS)基础上,通过引入双参数和...随着四元数的广泛应用,大型四元数结构矩阵方程的求解成为科学计算的重要课题。针对四元数亚正定系统AX=B,在新自共轭正定和斜自共轭分裂迭代(new positive definite and skew-self-conjugate splitting,NPSS)基础上,通过引入双参数和松弛加速技术,构建出两种新的混参分裂迭代格式,即非对称新自共轭正定和斜自共轭分裂迭代(asymmetric new positive definite and skew-self-conjugate splitting,ANPSS),以及超松弛非对称新自共轭正定和斜自共轭分裂迭代(successive over relaxation asymmetric new positive definite and skew-self-conjugate splitting,SANPSS),同时运用四元数矩阵特征值理论,证明了这两种迭代的收敛性,并给出相关参数的取值范围。采用四元数矩阵的复表示方法,在MATLAB环境下实现该系统的数值求解。数值算例表明,多参数的灵活选取,显示出所提混参分裂迭代相比NPSS迭代具有更高的收敛效率。展开更多
文摘This paper aims to present, in a unified manner, the algebraic techniques of eigen-problem which are valid on both the quaternions and split quaternions. This paper studies eigenvalues and eigenvectors of the v-quaternion matrices by means of the complex representation of the v-quaternion matrices, and derives an algebraic technique to find the eigenvalues and eigenvectors of v-quaternion matrices. This paper also gives a unification of algebraic techniques for eigenvalues and eigenvectors in quaternionic and split quaternionic mechanics.
文摘This paper aims to present, in a unified manner, algebraic techniques for linear equations which are valid on both the algebras of quaternions and split quaternions. This paper, introduces a concept of v-quaternion, studies the problem of v-quaternionic linear equations by means of a complex representation and a real representation of v-quaternion matrices, and gives two algebraic methods for solving v-quaternionic linear equations. This paper also gives a unification of algebraic techniques for quaternionic and split quaternionic linear equations in quaternionic and split quaternionic mechanics.
文摘In this paper, from the spacetime algebra associated with the Minkowski space ℝ3,1by means of a change of signature, we describe a quaternionic representation of the split-tetraquaternion algebra which incorporates the Pauli algebra, the split-biquaternion algebra and the split-quaternion algebra, we relate these algebras to Clifford algebras and we show the emergence of the stabilized Poincaré-Heisenberg algebra from the split-tetraquaternion algebra. We list without going into details some of their applications in Physics and in Born geometry.
文摘随着四元数的广泛应用,大型四元数结构矩阵方程的求解成为科学计算的重要课题。针对四元数亚正定系统AX=B,在新自共轭正定和斜自共轭分裂迭代(new positive definite and skew-self-conjugate splitting,NPSS)基础上,通过引入双参数和松弛加速技术,构建出两种新的混参分裂迭代格式,即非对称新自共轭正定和斜自共轭分裂迭代(asymmetric new positive definite and skew-self-conjugate splitting,ANPSS),以及超松弛非对称新自共轭正定和斜自共轭分裂迭代(successive over relaxation asymmetric new positive definite and skew-self-conjugate splitting,SANPSS),同时运用四元数矩阵特征值理论,证明了这两种迭代的收敛性,并给出相关参数的取值范围。采用四元数矩阵的复表示方法,在MATLAB环境下实现该系统的数值求解。数值算例表明,多参数的灵活选取,显示出所提混参分裂迭代相比NPSS迭代具有更高的收敛效率。
