{"paper":{"title":"A Necessary Condition for the Spectrum of Nonnegative Symmetric $ 5 \\times 5 $ Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Oren Spector, Raphael Loewy","submitted_at":"2016-02-16T15:14:05Z","abstract_excerpt":"Let $A$ be a nonnegative symmetric $ 5 \\times 5 $ matrix with eigenvalues $ \\lambda_1 \\geq \\lambda_2 \\geq \\lambda_3 \\geq \\lambda_4 \\geq \\lambda_5 $. We show that if $ \\sum_{i=1}^{5} \\lambda_{i} \\geq \\frac{1}{2} \\lambda_1 $ then $ \\lambda_3 \\leq \\sum_{i=1}^{5} \\lambda_{i} $. McDonald and Neumann showed that $ \\lambda_1 + \\lambda_3 + \\lambda_4 \\geq 0 $. Let $ \\sigma = \\left( \\lambda_1, \\lambda_2, \\lambda_3, \\lambda_4, \\lambda_5 \\right) $ be a list of decreasing real numbers satisfying:\n  1. $ \\sum_{i=1}^{5} \\lambda_{i} \\geq \\frac{1}{2} \\lambda_1 $,\n  2. $ \\lambda_3 \\leq \\sum_{i=1}^{5} \\lambda_{i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.05051","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":""},"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"}