Algebraic Techniques for Linear Equations over Quaternions and Split Quaternions: A Unified Approach in Quaternionic and Split Quaternionic Mechanics

Gang Wang,Tongsong Jiang,Zhenwei Guo,Dong Zhang

doi:10.4236/jamp.2019.78118

Abstract

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.

Highlights

IntroductionIn paper [2], the authors showed that a unit timelike quaternion represents a rotation in the Minkowski 3 space, and expressed Lorentzian rotation matrix generated
A quaternion, which was found in 1840 by William Rowan Hamilton [1], is in the form of q =q1 + q2i + q3 j + q4k, i2 = j2 = k2 = −1, ijk = −1, where q1, q2, q3, q4 ∈ R, and ij =− ji =k, jk =−kj =i, ki =−ik =j
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

Summary

Introduction

In paper [2], the authors showed that a unit timelike quaternion represents a rotation in the Minkowski 3 space, and expressed Lorentzian rotation matrix generated. A split quaternion (or coquaternion), which was found in 1849 by James Cockle [4], is in the form of q =q1 + q2i + q3 j + q4k , i2 = −1 , =j2 k=2 1 , ijk = 1 , where q1, q2 , q3, q4 ∈ R , and ij =− ji =k , jk =−kj =−i , ki =−ik =j and denotes the sets of quaternions and split quaternions respectively by H and Hs. The quaternion ring H and the split quaternion ring Hs are two associative and noncommutative 4-dimensional Clifford algebras, and the split quaternion ring Hs contains zero divisors, nilpotent elements and nontrivial idempotents. In paper [7], the authors studied eigenvalue problem of a rotation matrix in Minkowski 3 space by using split quaternions, and gave the characterizations of eigenvalues of a rotation matrix in Minkowski 3 space according to only first component of the corresponding quaternion. Quaternions and split quaternions in the study of geometry and physic are more than those, e.g

Objectives

Methods

Conclusion