A novel Moore–Penrose inverse to dual quaternion matrices with applications

In previous studies, the dual quaternion Moore–Penrose inverse was defined as a solution to the four Penrose equations, but it does not exist for all dual quaternion matrices. This paper introduces a novel dual quaternion Moore–Penrose (NDQMP) inverse by modifying the first Penrose equation $AXA=A$ to $AXA=A_e$, where $A_e$ is the essential approximation of $A$. The NDQMP inverse is shown to exist and be unique for all dual quaternion matrices. Using this inverse, we directly provide the necessary and sufficient conditions for the solvability of the dual quaternion matrix equation $AXB=C$, along with an expression for the least-squares solution. Finally, the main results are applied to kinematics and image processing, highlighting their practical utility.