{"paper":{"title":"Investigate Invertibility of Sparse Symmetric Matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Feng Wei","submitted_at":"2017-12-10T04:02:09Z","abstract_excerpt":"In this paper, we investigate the invertibility of sparse symmetric matrices. We show that for an $n\\times n$ sparse symmetric random matrix $A$ with $A_{ij} = \\delta_{ij} \\xi_{ij}$ is invertible with high probability. Here, $\\delta_{ij}$s, $i\\ge j$ are i.i.d. Bernoulli random variables with $\\mathbb{P} \\left(\\xi_{ij}=1 \\right) =p \\ge n^{-c}$, $\\xi_{ij}, i\\ge j$ are i.i.d. random variables with mean 0, variance 1 and finite forth moment $M_4$, and $c$ is constant depending on $M_4$. More precisely, $$ s_{\\rm min} (A) > \\varepsilon \\sqrt{\\frac{p}{n}}. $$ with high probability."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.04341","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"}