Generalized inverses of bordered matrices

Abstract

Several authors have considered nonsingular borderings A = B C D X  of B and investigated the properties of submatrices of A−1. Under specific conditions on the bordering, one can recover any g-inverse of B as a submatrix of A⁻¹. Borderings A of B are considered, where A might be singular, or even rectangular. If A is m × n and if B is an r × s submatrix of A, the consequences of the equality m + n − rank(A) = r + s − rank(B) with reference to the g-inverses of A are studied. It is shown that under this condition many properties enjoyed by nonsingular borderings have analogs for singular (or rectangular) borderings as well. We also consider g-inverses of the bordered matrix when certain rank additivity conditions are satisfied. It is shown that any g-inverse of B can be realized as a submatrix of a suitable g-inverse of A, under certain conditions.