{"paper":{"title":"The density of uncyclic matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Cheryl E. Praeger, S.P. Glasby","submitted_at":"2014-05-22T05:29:01Z","abstract_excerpt":"An element $X$ in the algebra ${\\rm M}(n,\\mathbb{F})$ of all $n\\times n$ matrices over a field $\\mathbb{F}$ is said to be $f$-cyclic if the underlying vector space considered as an $\\mathbb{F}[X]$-module has at least one cyclic primary component. These are the matrices considered to be `good' in the Holt-Rees version of Norton's irreducibility test in the MeatAxe algorithm. We prove that, for any finite field $\\mathbb{F}_q$, the proportion of matrices in ${\\rm M}(n,\\mathbb{F}_q)$ that are `not good' decays exponentially to zero as the dimension $n$ approaches infinity. Turning this around, we "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.5631","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"}