Solutions of the system of dual matrix equation AXB=B=BXA in two partial orders

Abstract

In this paper, we consider the solutions of the system of dual matrix equation $AXB=B=BXA$ in P-star partial order and D-star partial order, respectively. The solvability condition are obtained by applying the singular value decomposition (SVD) and the expression of general solutions to the dual matrix equations are provided. The special case in which $A=B$ is discussed and the properties of $\left\{1\right\}-$dual generalized inverse are used to verify the accuracy of the explicit solution of dual matrix equation $BXB=B$. Finally, the explicit formula for the unique minimum-norm solution is given.