{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:ADP734P37PKUYAFLP5FRPA2PYO","short_pith_number":"pith:ADP734P3","schema_version":"1.0","canonical_sha256":"00dffdf1fbfbd54c00ab7f4b17834fc3bfe0668b595812ce2e9ec706915157e0","source":{"kind":"arxiv","id":"1502.04189","version":2},"attestation_state":"computed","paper":{"title":"On the probability that all eigenvalues of Gaussian, Wishart, and double Wishart random matrices lie within an interval","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math-ph","math.IT","math.MP","stat.TH"],"primary_cat":"math.ST","authors_text":"Marco Chiani","submitted_at":"2015-02-14T11:05:22Z","abstract_excerpt":"We derive the probability that all eigenvalues of a random matrix $\\bf M$ lie within an arbitrary interval $[a,b]$, $\\psi(a,b)\\triangleq\\Pr\\{a\\leq\\lambda_{\\min}({\\bf M}), \\lambda_{\\max}({\\bf M})\\leq b\\}$, when $\\bf M$ is a real or complex finite dimensional Wishart, double Wishart, or Gaussian symmetric/Hermitian matrix. We give efficient recursive formulas allowing the exact evaluation of $\\psi(a,b)$ for Wishart matrices, even with large number of variates and degrees of freedom. We also prove that the probability that all eigenvalues are within the limiting spectral support (given by the Mar"},"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":"1502.04189","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2015-02-14T11:05:22Z","cross_cats_sorted":["cs.IT","math-ph","math.IT","math.MP","stat.TH"],"title_canon_sha256":"7e071600438eefe5300bd950dd79ebcdb76cd7cc531706af02d7a8912313d166","abstract_canon_sha256":"d0d9c87ec80f241b7c338f2ea5db01de5056955224d08063c2801a0abdde0c32"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:59.008161Z","signature_b64":"hztzYpha4kfZFyZ05UMNbuiqoir0htwxQ8Ps30Bp0IygfpMLkOZv05I7CGJBov9XvTZpGCOCWKVy1fSSGoi/Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"00dffdf1fbfbd54c00ab7f4b17834fc3bfe0668b595812ce2e9ec706915157e0","last_reissued_at":"2026-05-18T00:45:59.007682Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:59.007682Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the probability that all eigenvalues of Gaussian, Wishart, and double Wishart random matrices lie within an interval","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math-ph","math.IT","math.MP","stat.TH"],"primary_cat":"math.ST","authors_text":"Marco Chiani","submitted_at":"2015-02-14T11:05:22Z","abstract_excerpt":"We derive the probability that all eigenvalues of a random matrix $\\bf M$ lie within an arbitrary interval $[a,b]$, $\\psi(a,b)\\triangleq\\Pr\\{a\\leq\\lambda_{\\min}({\\bf M}), \\lambda_{\\max}({\\bf M})\\leq b\\}$, when $\\bf M$ is a real or complex finite dimensional Wishart, double Wishart, or Gaussian symmetric/Hermitian matrix. We give efficient recursive formulas allowing the exact evaluation of $\\psi(a,b)$ for Wishart matrices, even with large number of variates and degrees of freedom. We also prove that the probability that all eigenvalues are within the limiting spectral support (given by the Mar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.04189","kind":"arxiv","version":2},"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":"1502.04189","created_at":"2026-05-18T00:45:59.007756+00:00"},{"alias_kind":"arxiv_version","alias_value":"1502.04189v2","created_at":"2026-05-18T00:45:59.007756+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.04189","created_at":"2026-05-18T00:45:59.007756+00:00"},{"alias_kind":"pith_short_12","alias_value":"ADP734P37PKU","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_16","alias_value":"ADP734P37PKUYAFL","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_8","alias_value":"ADP734P3","created_at":"2026-05-18T12:29:10.953037+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/ADP734P37PKUYAFLP5FRPA2PYO","json":"https://pith.science/pith/ADP734P37PKUYAFLP5FRPA2PYO.json","graph_json":"https://pith.science/api/pith-number/ADP734P37PKUYAFLP5FRPA2PYO/graph.json","events_json":"https://pith.science/api/pith-number/ADP734P37PKUYAFLP5FRPA2PYO/events.json","paper":"https://pith.science/paper/ADP734P3"},"agent_actions":{"view_html":"https://pith.science/pith/ADP734P37PKUYAFLP5FRPA2PYO","download_json":"https://pith.science/pith/ADP734P37PKUYAFLP5FRPA2PYO.json","view_paper":"https://pith.science/paper/ADP734P3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1502.04189&json=true","fetch_graph":"https://pith.science/api/pith-number/ADP734P37PKUYAFLP5FRPA2PYO/graph.json","fetch_events":"https://pith.science/api/pith-number/ADP734P37PKUYAFLP5FRPA2PYO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ADP734P37PKUYAFLP5FRPA2PYO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ADP734P37PKUYAFLP5FRPA2PYO/action/storage_attestation","attest_author":"https://pith.science/pith/ADP734P37PKUYAFLP5FRPA2PYO/action/author_attestation","sign_citation":"https://pith.science/pith/ADP734P37PKUYAFLP5FRPA2PYO/action/citation_signature","submit_replication":"https://pith.science/pith/ADP734P37PKUYAFLP5FRPA2PYO/action/replication_record"}},"created_at":"2026-05-18T00:45:59.007756+00:00","updated_at":"2026-05-18T00:45:59.007756+00:00"}