On the commutative quaternion matrices

This study consists of five parts. The first part is an introduction devoted to the literature knowledge. In the second part of this study the fundamental definitions and theorems related to the complex numbers and complex matrices are given. In the third part, after the fundamental definitions and the theorems related to the elliptic number which is a general form of complex number are given, also, elliptic matrices are defined and studied. Some equivalence relations on elliptic matrices are given and relationships between the equivalence relations are investigated. Lastly, Sylvester-conjugate and Kalman-Yakubovich-conjugate linear equations are defined and studied on elliptic matrices. In the fourth part, after some algebraic properties of commutative quaternions are given, solutions of some linear equations and linear equations system of commutative quaternions are studied by means of real representation of commutative quaternions. Lastly, commutative quaternion matrices are defined and examined. Properties for elliptic matrices are investigated for commutative quaternions. In the fifth parth, after fundamental definitions and theorems of elliptic quaternions are given, The obtained results throughout the study are generalized for elliptic quaternions.