Generalization of the Sherman-Morrison-Woodbury formula involving the Schur complement

Let X∈Cm×m and Y∈Cn×n be nonsingular matrices, and let N∈Cm×n. Explicit expressions for the Moore-Penrose inverses of M=XNY and a two-by-two block matrix, under appropriate conditions, have been established by Castro-González et al. [Linear Algebra Appl. 471 (2015) 353-368]. Based on these results, we derive a novel expression for the Moore-Penrose inverse of A+UV* under suitable conditions, where A∈Cm×n,U∈Cm×r, and V∈Cn×r. In particular, if both A and I+V*A−1U are nonsingular matrices, our expression reduces to the celebrated Sherman-Morrison-Woodbury formula. Moreover, we extend our results to the bounded linear operators case.