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

TL;DR

This paper generalizes the Sherman-Morrison-Woodbury formula by deriving a new Moore-Penrose inverse expression for matrices involving the Schur complement, extending to bounded linear operators.

Contribution

It introduces a novel Moore-Penrose inverse formula for matrices with added rank-one updates, generalizing the Sherman-Morrison-Woodbury formula and extending to operator theory.

Findings

Derived a new Moore-Penrose inverse expression involving the Schur complement.

Reduced to Sherman-Morrison-Woodbury formula under specific nonsingularity conditions.

Extended results to bounded linear operators.

Abstract

Let $X\in\mathbb{C}^{m\times m}$ and $Y\in\mathbb{C}^{n\times n}$ be nonsingular matrices, and let $N\in\mathbb{C}^{m\times n}$. 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-Gonz\'{a}lez et al. [Linear Algebra Appl. 471 (2015) 353-368]. Based on these results, we derive a novel expression for the Moore-Penrose inverse of $A+UV^{\ast}$ under suitable conditions, where $A\in \mathbb{C}^{m\times n}$, $U\in \mathbb{C}^{m\times r}$, and $V\in \mathbb{C}^{n\times r}$. In particular, if both $A$ and $I+V^{\ast}A^{-1}U$ are nonsingular matrices, our expression reduces to the celebrated Sherman-Morrison-Woodbury formula. Moreover, we extend our results to the bounded linear operators case.