The generalized Moore-Penrose inverses of matrices over a ring with an involution * and unitary are discussed. Some necessary and sufficient conditions for existence of the generalized Moore-Penrose inverse of matrices over a ring are given. In particular, it is obtained that a necessary and sufficient condition for existence of the generalized Moore-Penrose inverse with relation to M and N of a matrix A over a ring is that A = GDH, where D2 = D, (MD)* = MD, (GD)*MGD + M(I-D) and DHN-1(DH)* + (I-D)M-1 are invertible.
Jian Miao CEN.
On Existence of the Generalized Moore-Penrose Inverses of Matrices over Ring. Acta Mathematica Sinica, Chinese Series, 2006, 49(3): 549-558 https://doi.org/10.12386/A2006sxxb0069