Generalized Inverses and Ranks of Block Matrices
Carl Dean Meyer · SIAM Journal on Applied Mathematics · 1973
For a completely general partitioned matrix of the form \[ M = [ {\begin{array}{*{20}c} A & C \\ R & D \\ \end{array} } ], \] a representation for (1)- and (1,2)-inverses of M is derived such that the known representations for some common partitioned matrices are special cases. The representation for a (1)-inverse of M is then used to obtain an expression for the rank of M in terms of ranks of matrices of lower order.