The minimal norm least squares Hermitian solution of the complex matrix equation AXB+CXD=E

In this paper, by applying the real representations of complex matrices, the particular structure of the real representations and the Moore–Penrose generalized inverse, we obtain the explicit expression of the minimal norm least squares Hermitian solution of the complex matrix equation $AXB+CXD=E$. And we also derive the minimal norm least squares Hermitian solution of the complex matrix equation $AXB=E$. Our proposed formulas only involve real matrices, and therefore are more effective and portable than those reported in Yuan and Liao (2014). The corresponding algorithms only perform real arithmetic which also consider the particular structure of the real representations of complex matrices. Two numerical examples are provided to demonstrate the effectiveness of our algorithms.