New characterizations of EP, generalized normal and generalized Hermitian elements in rings

The Moore-Penrose inverse in rings with involution

“…(1) ⇒ (10)- (13) It is easy to see that by Lemma 2.4. The equivalence between (10)-(13) can be seen by Lemma 2.9.…”

“…These conditions involve elements a, a * , a † , a # and also powers of these elements. In [13], more new characterizations of EP elements in rings are given by Mosić and Djordjević, which involve powers of their group and Moore-Penrose inverse. In [19], Tian and Wang presented some necessary and sufficient conditions such that A ∈ C n×n to be an EP matrix, which also involve powers of their group and Moore-Penrose inverse, where C n×n stands for the set of all n × n matrices over the field of complex numbers.…”

The class ofm-EPandm-normal matrices

“…However, we point out that in [11] m does not correspond necessarily to the index. In order to obtain a characterization for m-normal matrices, we take a matrix A ∈ C n×n of rank r > 0 in the Hartwig-Spindelböck decomposition, i.e., is 2-EP (to compute A † observe that the 2 × 2 sub-matrix in the N-W corner is nonsingular and the 2 × 2 sub-matrix in the S-E is J 2 (0)) but A is not 2-normal.…”

“…For some related results on m-normal matrices we refer the reader to [11] where some characterizations are given in the setting of rings. However, we point out that in [11] m does not correspond necessarily to the index.…”

On a new generalized inverse for matrices of an arbitrary index

“…Some properties for all these generalized inverses can be found in [2,3,4,7,8]. All of these generalized inverses are known to be used in important applications.…”