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We consider an ordered pair of linear transformations $A: V\\rightarrow V$ and $A^*: V\\rightarrow V$ that satisfy (i) and (ii) below.\n  (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal.\n  (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal.\n  We call such a pair a Leonard pair on $V$. Very roughly speaking, a Leonard pair is "},"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":"1308.3826","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-08-18T03:13:09Z","cross_cats_sorted":[],"title_canon_sha256":"14b10e4edadabb9c90e1c60a031d913377c1ea2ace7e36e8eaad5772ddb30150","abstract_canon_sha256":"d0dfd66634567285f8304f5dee4cba6b200b75bfd5da6bbc608f075df3c93f62"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:15:42.414545Z","signature_b64":"TI3Y+mf2Jekjk7FaHrhy0IOs2Z462lBYJ7xv3oTmh67AmCc4v4aRshe/uwAF2085/FUL9EmWFjpxh84JYP5qCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f41d00508f28b5499182e357e653189d4845f66d7dfed5c2701bb429f0726145","last_reissued_at":"2026-05-18T03:15:42.413875Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:15:42.413875Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A characterization of bipartite Leonard pairs using the notion of a tail","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Edward Hanson","submitted_at":"2013-08-18T03:13:09Z","abstract_excerpt":"Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\\rightarrow V$ and $A^*: V\\rightarrow V$ that satisfy (i) and (ii) below.\n  (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal.\n  (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal.\n  We call such a pair a Leonard pair on $V$. 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