Some Equivalence Relations and Results over the Commutative Quaternions and Their Matrices

In this paper, we give some equivalence relations and results over the commutative quaternions and their matrices. In this sense, consimilarity, semisimilarity, and consemisimilarity over the commutative quaternion algebra and commutative quaternion matrix algebra are established. Equalities of these equivalence relations are explicitly determined. Also Syvester-s-Conjugate commutative quaternion matrix equations are studied by means of real representation of the commutative quaternion matrices and consimilarity of the two commutative quaternion matrices.