Generalized Quaternions and Matrix Algebra

Symmetry, 2022

There are a total of 64 possible multiplication rules that can be defined starting with the generalized imaginary units first introduced by Hamilton. Of these sixty-four choices, only eight lead to non-commutative division algebras: two are associated to the left- and right-chirality quaternions, and the other six are generalizations of the split-quaternion concept first introduced by Cockle. We show that the 4×4 matrix representations of both the left- and right-chirality versions of the generalized split-quaternions are algebraically isomorphic and can be related to each other by 4×4 permutation matrices of the C2×C2 group. As examples of applications of the generalized quaternion concept, we first show that the left- and right-chirality quaternions can be used to describe Lorentz transformations with a constant velocity in an arbitrary spatial direction. Then, it is shown how each of the generalized split-quaternion algebras can be used to solve the problem of quantum-mechanical ...

2008

Since quaternions have isomorphic representations in matrix form we investigate various well known matrix decompositions for quaternions.

Computational Methods and Function Theory

In this article, we give the most general form of the quaternions algebra depending on 3-parameters. We define 3-parameter generalized quaternions (3PGQs) and study on various properties and applications. Firstly we present the definiton, the multiplication table and other properties of 3PGQs such as addition-substraction, multiplication and multiplication by scalar operations, unit and inverse elements, conjugate and norm. We give matrix representation and Hamilton operators for 3PGQs.We get polar represenation, De Moivre's and Euler's formulas with the matrix representations for 3PGQs. Besides, we give relations among the powers of the matrices associated with 3PGQs. Finally, Lie group and Lie algebra are studied and their matrix representations are shown. Also the Lie multiplication and the killing bilinear form are given.

Linear and Multilinear Algebra, 2011

This paper is a continuation of the article [28] 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 analyze the roots of the characteristic polynomials for 2 × 2 matrices. We establish a characterization for the set of left eigenvalues to intersect or be part of the boundary of the quaternion balls of Geršgorin.

2020

In order to study homogeneous system of linear differential equations, I considered vector space over division D-algebra and the theory of eigenvalues in non commutative division D-algebra. Since product in algebra is non-commutative, I considered two forms of product of matrices (section 2) and two forms of eigenvalues (section 4). In sections 5, 6, 7, I considered solving of homogenius system of differential equations.

WORLD SCIENTIFIC eBooks, 2010

Since quaternions have isomorphic representations in matrix form we investigate various well known matrix decompositions for quaternions.

Journal of the Egyptian Mathematical Society, 2013

In this paper, we establish the formulas of the extermal ranks of the quaternion matrix expression f(X 1 , X 2 ) = C 7 À A 4 X 1 B 4 À A 5 X 2 B 5 where X 1 , X 2 are variant quaternion matrices subject to quaternion matrix equations A 1 X 1 = C 1 , A 2 X 1 = C 2 , A 3 X 1 = C 3 , X 2 B 1 = C 4 , X 2 B 2 = C 5 , X 2 B 3 = C 6 . As applications, we give a new necessary and sufficient condition for the existence of solutions to some systems of quaternion matrix equations. Some results can be viewed as special cases of the results of this paper.

Mathematics, 2022

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

Mathematics, 2019

In this paper, we introduce the bicomplex generalized tribonacci quaternions. Furthermore, Binet’s formula, generating functions, and the summation formula for this type of quaternion are given. Lastly, as an application, we present the determinant of a special matrix, and we show that the determinant is equal to the n th term of the bicomplex generalized tribonacci quaternions.

Mathematical Intelligencer, 1996