On the Consimilarity of Split Quaternions and Split Quaternion Matrices

Abstract

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

The Moore-Penrose Inverses of Clifford Algebra $C\ell_{1,2}$

“…For example, Yildiz and Kosal etc. have studied comsimilarity and semisimilarity of split quaternions in [6,8]. In this section, we will introduce the concept of similarity of two elements in Cℓ 1,2 and obtain the necessary and sufficient conditions for them to be similar.…”

Similarity of hybrid numbers

“…5,6 Besides, we encounter the concepts of semisimilarity and con-semisimilarity as well as the similarity of quaternions in the literature. 1,[7][8][9][10] Two quaternions p and q are said to be semisimilar if there exist quaternions x and y satisfying equations xpy = q and ypx = q. Also, p and q are said to be con-semisimilar if there are x and y satisfying the equalities xpy = q and yqx = p. [11][12][13] The semisimilarity was first defined and studied by Hartwing and Bevis.…”

Algebraic techniques for least squares problem over generalized quaternion algebras: A unified approach in quaternionic and split quaternionic theory

“…After the work of Hamilton, in 1849, James Cockle introduced the split quaternion (or coquaternion) . Unlike the quaternions, it has been gradually recognized and studied, recently . In addition, some research on split quaternions are mainly related to geometric applications, for example, split quaternions are used to express screw motion in three‐dimensional Lorentzian space, the rotations in Minkowski 3‐space and so on .…”