Abstract

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.

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