This article establishes a few sufficient conditions of the forward-order law for core inverse elements in rings with involution. It also presents weighted and triple inverse. Additionally, we discuss hybrid involving different generalized inverses like Moore–Penrose inverse, group

We study the Moore–Penrose inverse (MP-inverse) in the setting of rings with involution. The results include the relation between regular, MPinvertible and well-supported elements. We present an algebraic proof of the reverse order rule for the MP-inverse valid under certain conditions on MP-invertible elements. Applications to C∗-algebras are given. 2000 Mathematics Subject Classification: 46L...

The concept of an inverse of a singular matrix seems to have been first introduced by Moore [1], [2] in 1920. Extensions of these ideas to general operators have been made by Tseng [3], [4], [5], but no systematic study of the subject was made until 1955 when Penrose [6], [7], unaware of the earlier work, redefined the Moore inverse in a slightly different way. About the same time one of the au...

A simple proof of the Greville formula for the recursive computation of the Moore-Penrose (MP) inverse of a matrix is presented. The proof utilizes no more than the elementary properties of the MP inverse.

Singular values and maximum rank minors of generalized inverses are studied. Proportionality of maximum rank minors is explained in terms of space equivalence. The Moore–Penrose inverse A† is characterized as the {1}–inverse of A with minimal volume.

In this paper, a class of Hessenberg matrices is presented for adoption as test matrices. The Moore-Penrose inverse and the Drazin inverse for each member of this class are determined explicitly.

The authors introduce a new type of matrix splitting generalizing the notion of B splitting and study its relationships with nonnegativity of the Moore-Penrose inverse and the group inverse. Mathematics subject classification (2010): 15A09, 15B48.

The reverse order rule (AB)† = B†A† for the Moore-Penrose inverse is established in several equivalent forms. Results related to other generalized inverses are also proved.