{"paper":{"title":"Schur's Lemma for Coupled Reducibility and Coupled Normality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Christian Jutten, Dana Lahat, Helene Shapiro","submitted_at":"2018-11-20T19:56:34Z","abstract_excerpt":"Let $\\mathcal A = \\{A_{ij} \\}_{i, j \\in \\mathcal I}$, where $\\mathcal I$ is an index set, be a doubly indexed family of matrices, where $A_{ij}$ is $n_i \\times n_j$. For each $i \\in \\mathcal I$, let $\\mathcal V_i$ be an $n_i$-dimensional vector space. We say $\\mathcal A$ is reducible in the coupled sense if there exist subspaces, $\\mathcal U_i \\subseteq \\mathcal V_i$, with $\\mathcal U_i \\neq \\{0\\}$ for at least one $i \\in \\mathcal I$, and $\\mathcal U_i \\neq \\mathcal V_i$ for at least one $i$, such that $A_{ij} (\\mathcal U_j) \\subseteq \\mathcal U_i$ for all $i, j$. Let $\\mathcal B = \\{B_{ij} \\}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.08467","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"}