{"paper":{"title":"Generalizing the Cauchy-Schwarz inequality: Hadamard powers and tensor products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["quant-ph"],"primary_cat":"math.FA","authors_text":"Charles Torrance, Luis M. B. Varona, Nathaniel Johnston, Sarah Plosker","submitted_at":"2025-07-14T14:36:34Z","abstract_excerpt":"We explore and generalize a Cauchy-Schwarz-type inequality originally proved in [Electronic Journal of Linear Algebra 35, 156-180 (2019)]: $\\|\\mathbf{v}^2\\|\\|\\mathbf{w}^2\\| - \\langle\\mathbf{v}^2,\\mathbf{w}^2\\rangle \\leq \\|\\mathbf{v}\\|^2\\|\\mathbf{w}\\|^2 - \\langle\\mathbf{v},\\mathbf{w}\\rangle^2$ for all $\\mathbf{v},\\mathbf{w} \\in \\mathbb{R}^n$. We present three new proofs of this inequality that better illustrate \"why\" it is true and generalize it in several different ways: we generalize from vectors to matrices, we explore which exponents other than 2 result in the inequality holding, and we der"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.10327","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/2507.10327/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"}