{"paper":{"title":"Generalization of the Sherman-Morrison-Woodbury formula involving the Schur complement","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Xuefeng Xu","submitted_at":"2016-07-06T12:02:56Z","abstract_excerpt":"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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.01579","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}