{"paper":{"title":"The smallest singular value of inhomogeneous square random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.MG"],"primary_cat":"math.PR","authors_text":"Galyna V. Livshyts, Konstantin Tikhomirov, Roman Vershynin","submitted_at":"2019-09-10T01:07:46Z","abstract_excerpt":"We show that for an $n\\times n$ random matrix $A$ with independent uniformly anti-concentrated entries, such that $\\mathbb{E} ||A||^2_{HS}\\leq K n^2$, the smallest singular value $\\sigma_n(A)$ of $A$ satisfies $$ P\\left( \\sigma_n(A)\\leq \\frac{\\varepsilon}{\\sqrt{n}} \\right) \\leq C\\varepsilon+2e^{-cn},\\quad \\varepsilon \\ge 0. $$ This extends earlier results of Rudelson and Vershynin, and Rebrova and Tikhomirov by removing the assumption of mean zero and identical distribution of the entries across the matrix, as well as the recent result of Livshyts, where the matrix was required to have i.i.d. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.04219","kind":"arxiv","version":4},"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/1909.04219/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"}