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

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  • Xu, Xuefeng

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.

Suggested Citation

  • Xu, Xuefeng, 2017. "Generalization of the Sherman–Morrison–Woodbury formula involving the Schur complement," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 183-191.
  • Handle: RePEc:eee:apmaco:v:309:y:2017:i:c:p:183-191
    DOI: 10.1016/j.amc.2017.03.039
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    Cited by:

    1. Xu, Xuefeng, 2020. "On the perturbation of the Moore–Penrose inverse of a matrix," Applied Mathematics and Computation, Elsevier, vol. 374(C).

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