The Moore-Penrose inverse of a partitioned nonnegative definite matrix

Consider an arbitrary symmetric nonnegative definite matrix A and its Moore-Penrose inverse A(+), partitioned, respectively as A = ((E)(F') (F)(H)) and A(+) = ((Gt)(G2)(G2')(G4)). Explicit expressions for G(1), G(2) and G(4) in terms of E, F and H are given. Moreover, it is proved that the generalized Schur complement (A(+)/G(4)) = G(1) - G(2)G(4)(+)G'(2) is always below the Moore-Penrose inverse (A/H)(+) of the generalized Schur complement (A/H) = E - FH+F' with respect to the Lowner partial ordering. (C) 2000 Elsevier Science Inc. All rights reserved. AMS classification: 15A09.