RANK OF A DIFFERENCE OF MATRICES AND ASSOCIATED GENERALIZED INVERSES

Various representations are given to characterize the rank of A-S in terms of rank A+k where A and S are arbitrary complex matrices and k is a function of A and S. It is shown that if S=AMA for some matrix M, and if G is any matrix satisfying A=AGA, then rank(A-S) = rankA-nullity (I-SG). Several alternative forms of this result are established, as are many equivalent conditions to have rank(A-S) = rankA-rankS. General forms for the Moore-Penrose inverse of matrices A-S are developed which include as special cases various results by Penrose, Wedin, Hartwig and others. © 1979.