Construction of EPr generalized inverses by inversion of nonsingular matrices

Abstract

Any matrix 8 s uc h that A8A = A is ca ll ed a C,·inve rse of A and a C,-inverse of A such that BA8 = B is ca ll ed a C,-inverse of A. Some properties of such inverses are es tabli s hed. It is shown that if A is p-square of ra nk q < p and P is any pos itive semide finit e matrix , whose rank is the nullity of A, such that U = A + Pis nonsin gular , then B = UIAUI is a C, ·inverse of A with the property that null space 8 = null s pace 8 *. That s uc h a P exis ts for a rbitrary squ are A is shown. The relation be tween thi s res ult and th e work of Go ldm an a nd Zelen is discussed.