{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:XZHSBLP77RG4E6VRDCCAYX7JXM","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"2d7c0b76f45260677337e92bdd10d8a14668feca9103c6f38b22130dc82b339d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-10-10T19:22:22Z","title_canon_sha256":"2a44ca0d1e5017f9489123c83ca8bc3fcf6283c20e9014487ad5fede3656b8e0"},"schema_version":"1.0","source":{"id":"1210.3009","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.3009","created_at":"2026-05-18T03:43:30Z"},{"alias_kind":"arxiv_version","alias_value":"1210.3009v1","created_at":"2026-05-18T03:43:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.3009","created_at":"2026-05-18T03:43:30Z"},{"alias_kind":"pith_short_12","alias_value":"XZHSBLP77RG4","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XZHSBLP77RG4E6VR","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XZHSBLP7","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:c316faf062da755a05a7672cdce8b1d9057bc4a3593b85558bd38f2a227eece7","target":"graph","created_at":"2026-05-18T03:43:30Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"It is known that a $2\\times 2$ quaternionic matrix has one, two or an infinite number of left eigenvalues, but the available algebraic proofs are difficult to generalize to higher orders. In this paper a different point of view is adopted by computing the topological degree of a characteristic map associated to the matrix and discussing the rank of the differential. The same techniques are extended to $3\\times 3$ matrices, which are still lacking a complete classification.","authors_text":"E. Mac\\'ias-Virg\\'os, M. J. Pereira-S\\'aez","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-10-10T19:22:22Z","title":"A topological approach to left eigenvalues of quaternionic matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.3009","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:3e483ba3463eb8d660e008b0c568ec2a7b3f916070faba598d275da97c15a047","target":"record","created_at":"2026-05-18T03:43:30Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"2d7c0b76f45260677337e92bdd10d8a14668feca9103c6f38b22130dc82b339d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-10-10T19:22:22Z","title_canon_sha256":"2a44ca0d1e5017f9489123c83ca8bc3fcf6283c20e9014487ad5fede3656b8e0"},"schema_version":"1.0","source":{"id":"1210.3009","kind":"arxiv","version":1}},"canonical_sha256":"be4f20adfffc4dc27ab118840c5fe9bb37069622508bd9d143c5f02ed312c0f3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"be4f20adfffc4dc27ab118840c5fe9bb37069622508bd9d143c5f02ed312c0f3","first_computed_at":"2026-05-18T03:43:30.821850Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:43:30.821850Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/OK6/xQpu5Eb8jbHLCke7043897b79XSd++ibS+nPzQ/25rA65OkknHV1LmFZmtvbLMn8js0Hnz+TbAl4cR5CA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:43:30.822521Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.3009","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e483ba3463eb8d660e008b0c568ec2a7b3f916070faba598d275da97c15a047","sha256:c316faf062da755a05a7672cdce8b1d9057bc4a3593b85558bd38f2a227eece7"],"state_sha256":"5afe178389578908aaebae54a46bd1888004049dbec0d4d4f0bd59c8e286d792"}