In this paper we exhibit, under suitable conditions, a neat relationship between the Moore–Penrose generalized inverse of a sum of two matrices and the Moore–Penrose generalized inverses of the individual terms. We include an application to the parallel sum of matrices. AMS 1991 subject classifications. Primary 15A09; secondary 15A18.

An iterative algorithm for estimating the Moore-Penrose generalized inverse is developed. The main motive for the construction of algorithm is simultaneous usage of Penrose equations (2) and (4). Convergence properties of the introduced method are considered as well as their first-order and the second-order error terms. Numerical experience is also presented. AMS Subj. Class.: 15A09.

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.