Some Results on Generalized Inverses

Abstract

B = S0B + (I S?S)Y, where Y is an arbitrary matrix of appropriate dimensions. The matrix S0 is called a generalized inverse of S and has the property SSgs = S. Four types of generalized inverses can be defined as follows: DEFINITION 1. Sg is said to be a generalized inverse of S if (1.1) SSgS = S. DEFINITION 2. Sr is said to be a reflexive generalized inverse of S if (1.2) SSrS = S and SrSSr = Sr. DEFINITION 3. Sn is said to be a normalized generalized inverse of S if (1.3) SS'S = , S) SS = S', and (SS')* = SS'. DEFINITION 4. St is said to be the pseudoinverse of S if