Unified SMW-Like Identities of Low-Rank Updates for Generalized Inverses and Pseudoinverses

. We present new low-rank-update identities for generalized inverses and pseudoinverses of rectangular matrices, unifying previous generalizations of the Sherman–Morrison–Woodbury (SMW) identity and Riedel's rank-augmentation formulas. First, we establish generalized SMW identities for \ (\1\\) -inverses and pseudoinverses under less restrictive conditions when the matrix rank is preserved. Second, we further generalize Riedel's formulas for rank augmentation to pseudoinverses of rectangular matrices and to specific classes of generalized inverses (namely, those involving \ (\1, 3\\) - and \ (\1, 4\\) -inverses) when matrix ranges are altered by the update. Finally, we introduce unified low-rank-update identities that encompass both cases. These identities retain SMW's efficiency, extend its applicability to rectangular matrices, and offer new tools for updating generalized inverses in various applications. Keywordsgeneralized inverseMoore–Penrose pseudoinverseSherman–Morrison–Woodbury identitylow-rank updaterank-deficient least squaresrank augmentationMSC codes15A0915A2365F0565F20