{"paper":{"title":"On the structure of the fundamental subspaces of acyclic matrices with $0$ in the diagonal","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adri\\'an Pastine, Daniel A. Jaume","submitted_at":"2018-02-27T20:02:48Z","abstract_excerpt":"A matrix is called acyclic if replacing the diagonal entries with $0$, and the nonzero diagonal entries with $1$, yields the adjacency matrix of a forest. In this paper we show that null space and the rank of a acyclic matrix with $0$ in the diagonal is obtained from the null space and the rank of the adjacency matrix of the forest by multipliying by non-singular diagonal matrices. We combine these methods with an algorithm for finding a sparsest basis of the null space of a forest to provide an optimal time algorithm for finding a sparsest basis of the null space of acyclic matrices with $0$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.10142","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"}