{"paper":{"title":"On the smallest singular value of the product of random and deterministic matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Achintya Raya Polavarapu, Brayden Letwin","submitted_at":"2026-07-07T20:31:42Z","abstract_excerpt":"Let $A=(a_{ij})$ be an $n\\times n$ real-valued random matrix with independent, mean-zero, variance-one entries whose fourth moments are uniformly at most $K$. Suppose that there exists $\\kappa \\in (0, 1)$ such that the entries of $A$ satisfy $$ \\max_{i,j}\\sup_{u \\in \\mathbb{R}} \\mathbb{P}(\\lvert a_{ij} - u\\rvert < 1) \\le \\kappa. $$ We prove that there are constants $c,C>0$, depending only on $K$ and $\\kappa$, such that for every fixed invertible $n\\times n$ matrix $M$ and every $\\varepsilon\\ge0$, $$ \\mathbb{P}!\\left(s_{\\min}(MA) \\le \\frac{\\varepsilon}{\\lVert M^{-1}\\rVert_{\\mathrm{HS}}}\\right) "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.06785","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.06785/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}