Matrix Decompositions Involving the Schur Complement
David H Carlson · SIAM Journal on Applied Mathematics · 1975
Necessary and sufficient conditions are given for the rank of a sum of matrices over an arbitrary field to equal the sum of the ranks of the matrices. Several decompositions are given of a partitioned matrix into a sum of matrices. These provide a unified treatment of some classical results and some recent results on the ranks and generalized inverses of partitioned matrices, and lead to an abstract characterization of a concept related to the previously studied Schur complement.