Some properties of complex quaternion and complex split quaternion matrices

Abstract

The aim of this study is to investigate some properties of complex quaternion and complex split quaternion matrices. To verify this, we use 2x2 complex matrix representation of these quaternions. Moreover, we present a method to find the determinant of complex quaternion and complex split quaternion matrices. Finally, we research some special matrices for quaternions above.

A Unified Approach: Split Quaternions with Quaternion Coefficients and Quaternions with Dual Coefficients

“…Here i, j, k denote the quaternion units and commutes with e 1 , e 2 and e 3 , respectively. Furthermore, S P = P 0 E 0 is the scalar part and V P = P 1 E 1 + P 2 E 2 + P 3 E 3 is vector part of P in [11]. The set of split quaternions with quaternion coefficients are denoted by H Q S in [11].…”

“…Furthermore, S P = P 0 E 0 is the scalar part and V P = P 1 E 1 + P 2 E 2 + P 3 E 3 is vector part of P in [11]. The set of split quaternions with quaternion coefficients are denoted by H Q S in [11]. Basis elements of 4x4 matrices are given as follows:…”

“…The conjugate, quaternionic conjugate and total conjugate are defined, respectively, as follows [11]:…”

“…where p 11 = P 0 + iP 1 and p 12 = P 2 + iP 3 are considered for calculations. If P = 1, then P is called unit split quaternion with quaternion coefficients in [11].…”

“…Some algebraic concepts for complex quaternions and complex split quaternions were given in [11,12]. In these studies, a 4 × 4 quaternion coefficients matrix representation was used.…”