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We obtain the optimal containment region for the generalized eigenvalue $\\lambda$ satisfying $$\\lambda \\pmatrix A_{11} & 0 \\cr 0 & A_{22}\\cr\\pmatrix x = \\pmatrix 0 & A_{12} \\cr A_{21} & 0\\cr\\pmatrix x \\quad \\hbox{for some nonzero} x \\in \\IC^n,$$ and the optimal eigenvalue containment region of the matr"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1306.4916","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2013-06-20T15:41:00Z","cross_cats_sorted":[],"title_canon_sha256":"2922187ead355e8bd7c223c9e5375ae75a6a953a3b13156e832f60ca6bf670a3","abstract_canon_sha256":"024ea7abb9d608db9b49c59250e2dc612477dedcc5cc4826f1fdf7971ea0e28a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:15:33.872715Z","signature_b64":"0IXu4vI7MM/KmYCgNUTuiJUdEVKdvORgfcbwXQb9+sOjPVzdUbkAP7p+rIEhlIbLMpxAcqcufnUF/eU8rckDAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4c4c29a2c6c16254b3e185137cd11ec6423b7a63c94d32e3d4a639f43921224e","last_reissued_at":"2026-05-18T03:15:33.872182Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:15:33.872182Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Determinantal and eigenvalue inequalities for matrices with numerical ranges in a sector","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Chi-Kwong Li, Nung-sing Sze","submitted_at":"2013-06-20T15:41:00Z","abstract_excerpt":"Let $A = \\pmatrix A_{11} & A_{12} \\cr A_{21} & A_{22}\\cr\\pmatrix \\in M_n$, where $A_{11} \\in M_m$ with $m \\le n/2$, be such that the numerical range of $A$ lies in the set $\\{e^{i\\varphi} z \\in \\IC: |\\Im z| \\le (\\Re z) \\tan \\alpha\\}$, for some $\\varphi \\in [0, 2\\pi)$ and $\\alpha \\in [0, \\pi/2)$. We obtain the optimal containment region for the generalized eigenvalue $\\lambda$ satisfying $$\\lambda \\pmatrix A_{11} & 0 \\cr 0 & A_{22}\\cr\\pmatrix x = \\pmatrix 0 & A_{12} \\cr A_{21} & 0\\cr\\pmatrix x \\quad \\hbox{for some nonzero} x \\in \\IC^n,$$ and the optimal eigenvalue containment region of the matr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.4916","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1306.4916","created_at":"2026-05-18T03:15:33.872266+00:00"},{"alias_kind":"arxiv_version","alias_value":"1306.4916v3","created_at":"2026-05-18T03:15:33.872266+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.4916","created_at":"2026-05-18T03:15:33.872266+00:00"},{"alias_kind":"pith_short_12","alias_value":"JRGCTIWGYFRF","created_at":"2026-05-18T12:27:49.015174+00:00"},{"alias_kind":"pith_short_16","alias_value":"JRGCTIWGYFRFJM7B","created_at":"2026-05-18T12:27:49.015174+00:00"},{"alias_kind":"pith_short_8","alias_value":"JRGCTIWG","created_at":"2026-05-18T12:27:49.015174+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JRGCTIWGYFRFJM7BQUJXZUI6YZ","json":"https://pith.science/pith/JRGCTIWGYFRFJM7BQUJXZUI6YZ.json","graph_json":"https://pith.science/api/pith-number/JRGCTIWGYFRFJM7BQUJXZUI6YZ/graph.json","events_json":"https://pith.science/api/pith-number/JRGCTIWGYFRFJM7BQUJXZUI6YZ/events.json","paper":"https://pith.science/paper/JRGCTIWG"},"agent_actions":{"view_html":"https://pith.science/pith/JRGCTIWGYFRFJM7BQUJXZUI6YZ","download_json":"https://pith.science/pith/JRGCTIWGYFRFJM7BQUJXZUI6YZ.json","view_paper":"https://pith.science/paper/JRGCTIWG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1306.4916&json=true","fetch_graph":"https://pith.science/api/pith-number/JRGCTIWGYFRFJM7BQUJXZUI6YZ/graph.json","fetch_events":"https://pith.science/api/pith-number/JRGCTIWGYFRFJM7BQUJXZUI6YZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JRGCTIWGYFRFJM7BQUJXZUI6YZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JRGCTIWGYFRFJM7BQUJXZUI6YZ/action/storage_attestation","attest_author":"https://pith.science/pith/JRGCTIWGYFRFJM7BQUJXZUI6YZ/action/author_attestation","sign_citation":"https://pith.science/pith/JRGCTIWGYFRFJM7BQUJXZUI6YZ/action/citation_signature","submit_replication":"https://pith.science/pith/JRGCTIWGYFRFJM7BQUJXZUI6YZ/action/replication_record"}},"created_at":"2026-05-18T03:15:33.872266+00:00","updated_at":"2026-05-18T03:15:33.872266+00:00"}