{"paper":{"title":"Commutators, matrices and an identity of Copeland","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RA","authors_text":"Darij Grinberg","submitted_at":"2019-08-24T18:12:08Z","abstract_excerpt":"Given two elements $a$ and $b$ of a noncommutative ring, we express $\\left( ba\\right)^n$ as a \"row vector times matrix times column vector\" product, where the matrix is the $n$-th power of a matrix with entries $\\dbinom{i}{j}\\operatorname{ad}_a^{i-j}\\left( b\\right)$. This generalizes a formula by Tom Copeland used in the study of Pascal-style matrices."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.09179","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/1908.09179/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"}