The General Solution of Quaternion Matrix Equation Having η-Skew-Hermicity and Its Cramer’s Rule

Abstract

We determine some necessary and sufficient conditions for the existence of the η-skew-Hermitian solution to the following system AX-(AX)η⁎+BYBη⁎+CZCη⁎=D,Y=-Yη⁎,Z=-Zη⁎ over the quaternion skew field and provide an explicit expression of its general solution. Within the framework of the theory of quaternion row-column noncommutative determinants, we derive its explicit determinantal representation formulas that are an analog of Cramer’s rule. A numerical example is also provided to establish the main result.

Solvability Conditions and General Solution of a System of Matrix Equations Involving η-Skew-Hermitian Quaternion Matrices

“…Now, we consider a few special cases of linear system (5). We remark that, if we consider the corresponding matrices to be equal to zero in (5), then we obtain the general solution of (3), which is the main result of [48], that is, Theorem 1 recovers Lemma 6.…”