In this study, we introduce the concept of commutative quaternions and commutative quaternion matrices. Firstly, we give some properties of commutative quaternions and their fundamental matrices. After that we investigate commutative quaternion matrices using properties of complex matrices. Then we define the complex adjoint matrix of commutative quaternion matrices and give some of their properties.

In this study, we derive the expressions of the minimal norm least-squares solution for the reduced biquaternion (RB) matrix equation AX=B by using the $e_1-e_2$ form of RB matrices, and the Moore–Penrose generalized inverse. Moreover, we investigate their applications in colour image restoration. Therefore, many useful image restoration methods can be extended to reduced biquaternion algebra without separating the colour image into three channels. To prove the authenticity of our results and to distinguish them from existing ones, some illustrative examples are also given.

In this work, we have established universal similarity factorization equalities over the commutative quaternions and their matrices. Based on these equalities, real matrix representations of commutative quaternions and their matrices have been derived, and their algebraic properties and fundamental equations have been determined. Moreover, illustrative examples are provided to support our results.

In this paper, we introduce the concept of consimilarity of split quaternions and split quaternion matrices. In his regard, we examine the solvability conditions and general solutions of the equations and in split quaternions and split quaternion matrices, respectively. Moreover, coneigenvalue and coneigenvector are defined for split quaternion matrices. Some consequences are also presented.