Further Results on Generalized Inverses of Partitioned Matrices
Frank J. Hall, Robert E. Hartwig · SIAM Journal on Applied Mathematics · 1976
Necessary and sufficient conditions are given for the blocks in generalized inverses of the partitioned matrices $\left[ {\begin{array}{*{20}c} A & B \\ \end{array} } \right]$ and $[ {\begin{array}{*{20}c} A \\ C \\ \end{array} } ]$ to be independent of each other. Some of these conditions are then incorporated into conditions suitable for finding the Moore–Penrose inverse of a particular bordered matrix.