MATHEMATICAL METHODS IN THE APPLIED SCIENCES, vol.42, no.16, pp.5215-5232, 2019 (Journal Indexed in SCI)
The split and hyperbolic (countercomplex) octonions are eight-dimensional nonassociative algebras over the real numbers, which are in the form X = x(0)e(0) + Sigma(7)(j=1)x(j)e(j) e(m)'s (0 <= m <= 7) have different properties for them. The main purpose of this paper is to define the split-type octonion and its matrix whose inputs are split-type octonions and give some properties for them by using the real quaternions, split, and hyperbolic (countercomplex) octonions. On the other hand, to make some definitions, we present some operations on the split-type octonions. Also, we show that every split-type octonions can be represented by 2 x 2 real quaternion matrix and 4 x 4 complex number matrix. The information about the determinants of these matrix representations is also given. Besides, the main features of split-type octonion matrix concept are given by using properties of real quaternion matrices. Then, 8n x 8nreal matrix representations of split-type octonion matrices are shown, and some algebraic structures are examined. Additionally, we introduce real quaternion adjoint matrices of split-type octonion matrices. Moreover, necessary and sufficient conditions and definitions are given for split-type octonion matrices to be special split-type octonion matrices. We describe some special split-type octonion matrices. Finally, oct-determinant of split-type octonion matrices is defined. Definitive and understandable examples of all definitions, theorems, and conclusions were given for a better understanding of all these concepts.