GENERALIZED SCHUR COMPLEMENTS

Let A be an n×n complex matrix. For a suitable subspace M of Cn the Schur compression A M and the (generalized) Schur complement A/M are defined. If A is written in the form A= B C S T according to the decomposition Cn=M⊕M⊥ and if B is invertible, then AM= B C S SB-1C and A/M= 0 0 0 T-SB-1C· The commutativity rule for Schur complements is proved: (A/M)/N=(A)/N)/M· This unifies Crabtree and Haynsworth's quotient formula for (classical) Schur complements and Anderson's commutativity rule for shorted operators. Further, the absorption rule for Schur compressions is proved: (A/M)N=(AN)M=AM whenever M⊆N. © 1979.