{"paper":{"title":"Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Andrii Dmytryshyn, Bo K\\r{a}gstr\\\"om, Lena Klimenko, Vladimir V. Sergeichuk, Vyacheslav Futorny","submitted_at":"2010-04-20T22:13:25Z","abstract_excerpt":"We construct the Hasse diagrams $G_2$ and $G_3$ for the closure ordering on the sets of congruence classes of $2\\times 2$ and $3\\times 3$ complex matrices. In other words, we construct two directed graphs whose vertices are $2\\times 2$ or, respectively, $3\\times 3$ canonical matrices under congruence and there is a directed path from $A$ to $B$ if and only if $A$ can be transformed by an arbitrarily small perturbation to a matrix that is congruent to $B$.\n  A bundle of matrices under congruence is defined as a set of square matrices $A$ for which the pencils $A+\\lambda A^T$ belong to the same "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.3590","kind":"arxiv","version":3},"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"}