{"paper":{"title":"An arithmetic-geometric mean inequality for products of three matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Arie Israel, Felix Krahmer, Rachel Ward","submitted_at":"2014-11-02T23:14:55Z","abstract_excerpt":"Consider the following noncommutative arithmetic-geometric mean inequality: given positive-semidefinite matrices $\\mathbf{A}_1, \\dots, \\mathbf{A}_n$, the following holds for each integer $m \\leq n$: $$ \\frac{1}{n^m}\\sum_{j_1, j_2, \\dots, j_m = 1}^{n} ||| \\mathbf{A}_{j_1} \\mathbf{A}_{j_2} \\dots \\mathbf{A}_{j_m} ||| \\geq \\frac{(n-m)!}{n!} \\sum_{\\substack{j_1, j_2, \\dots, j_m = 1 \\\\ \\text{all distinct}}}^{n} ||| \\mathbf{A}_{j_1} \\mathbf{A}_{j_2} \\dots \\mathbf{A}_{j_m} |||,$$ where $||| \\cdot |||$ denotes a unitarily invariant norm, including the operator norm and Schatten p-norms as special cases"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.0333","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":""},"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"}