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

Xuefeng Xu

Applied Mathematics and Computation, 2017, vol. 309, issue C, 183-191

Abstract: 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.

Keywords: Sherman–Morrison–Woodbury formula; Moore–Penrose inverse; Schur complement (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:309:y:2017:i:c:p:183-191

DOI: 10.1016/j.amc.2017.03.039

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