Characterizations of EP, normal, and Hermitian matrices

Various characterizations of EP, normal, and Hermitian matrices are obtained by exploiting an elegant representation of matrices derived by Hartwig and Spindelböck [7 Hartwig, RE and Spindelböck, K. 1984. Matrices for which A* and A† commute. Linear and Multilinear Algebra, 14: 241–256. [Taylor & Francis Online] , [Google Scholar], Corollary 6]. One aim of the present article is to demonstrate its usefulness when investigating different matrix identities. The second aim is to extend and generalize lists of characterizations of Equal Projectors (EP), normal, and Hermitian matrices known in the literature, by providing numerous sets of equivalent conditions referring to the notions of conjugate transpose, Moore–Penrose inverse, and group inverse.