An Efficient Method for Solving Equations in Generalized Quaternion and Octonion Algebras

Abstract

Quaternions often appear in wide areas of applied science and engineering such as wireless communications systems, mechanics, etc. It is known that are two types of non-isomorphic generalized quaternion algebras, namely: the algebra of quaternions and the algebra of coquaternions. In this paper, we present the formulae to pass from a basis in the generalized quaternion algebras to a basis in the division quaternions algebra or to a basis in the coquaternions algebra and vice versa. The same result was obtained for the generalized octonion algebra. Moreover, we emphasize the applications of these results to the algebraic equations and De Moivre's formula in generalized quaternion algebras and in generalized octonion division algebras.

New Insight into Quaternions and Their Matrices

“…The questions about numbers, hypercomplex numbers and quaternions included questions about their matrices. Inspired by matrix forms in the study [23], we give an answer for the question of the alternative representation of generalized quaternion matrix with elliptic number entries (see elliptic biquaternions in [40]). So this matrix is in the form:…”

New Curiosity Bivariate Quadratic Quaternionic Polynomials and Their Roots

“…In [15], the solutions of the linear quaternion equation are considered. In [5], they study some equations in generalized octonion and quaternion algebras.…”

On the Para-Octonions; A Non-Associative Normed algebra

“…In this section, we give a brief summary of the generalized octonions. For detailed information about these octonions, we refer the reader to [1].…”

“…In this paper, we study some algebraic properties of para-octonions, which is called 1 8 − octonions in [9]. A pare-octonions can be written in form a dual quasi-quaternions.…”