Characterizations of normal matrices using the group inverse and the Moore–Penrose inverse

This paper presents a systematic study of the characterization of normal matrices through the properties of products involving their generalized inverses. A series of necessary and sufficient conditions characterizing normal matrices has been established from multiple perspectives, including one-sided Hermitian nature, one-sided equalities, X -commutativity, classical commutativity conditions, the existence of solutions to associated matrix equations, and the explicit forms of the general solutions to these equations.