Characterization of generalized inverses by a rank equation

摘要

If A is a nonsingular matrix of order n and if B=C=In, then the inverse of A is the unique matrix X such thatrankABCX=rank(A).In this paper, we generalize this fact to any matrix A of dimension m×n over the complex field to obtain analogous results for outer inverses of A. The converse problem is also considered in the sense that B and C are characterized when Ad, A♯, A(1,2), A(1,2,3) and A(1,2,4) are solutions to this equation, respectively. This contributes to certain recent results in the literature, including that obtained by Groß [Linear Algebra Appl. 289 (1999) 127].