The Right–Left WG Inverse Solutions to Quaternion Matrix Equations

This paper studies new characterizations and expressions of the weak group (WG) inverse and its dual over the quaternion skew field. We introduce a dual to the weak group inverse for the first time in the literature and give some new characterizations for both the WG inverse and its dual, named the right and left weak group inverses for quaternion matrices. In particular, determinantal representations of the right and left WG inverses are given as direct methods for their constructions. Our other results are related to solving the two-sided constrained quaternion matrix equation AXB=C and the according approximation problem that could be expressed in terms of the right and left WG inverse solutions. Within the framework of the theory of noncommutative row–column determinants, we derive Cramer’s rules for computing these solutions based on determinantal representations of the right and left WG inverses. A numerical example is given to illustrate the gained results.