Relationships between generalized inverses of a matrix and generalized inverses of its rank-one-modifications Original Research Article

For given complex matrix A and nonzero complex vectors b, c, relationships between generalized inverses of A and generalized inverses of the rank-one-modified matrix M=A+bc* (with c* being the conjugate transpose of c) are investigated. The following three questions are considered: (i) when a given generalized inverse A− belongs to the set M{1} of all generalized inverses of M, (ii) when does A−set membership, variantA{1} exist such that simultaneously A−set membership, variantM{1}, and (iii) when the set A{1} is a subset of M{1}. The same questions are also discussed for reflexive generalized inverses of A and M. The answers obtained are commented from the view-point of a result concerning comparison of ranks of M and A.