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We consider a pair of linear transformations $A : V \\to V$ and $A^* : V \\to V$ 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 {\\em Leonard pair} on $V$. Let $v_0,v_1,...,v_d$ (resp. $w_0,w_1,...,w_d$) denote a"},"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":"math/0608623","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.RA","submitted_at":"2006-08-25T00:27:39Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"4df9341a20d02754de9de55e9430b2541c27b8f939e29f2e828e4146b76819ce","abstract_canon_sha256":"ed7a3a213be4da2e05a168c272ded143d2e985530578e059545e7f625d4b57cb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:56:09.085037Z","signature_b64":"UcO+peb+AdRy9xyP7M07rlOUY4RcZ+RMlC1biMDNEDW+AEDVDsGC4LNplFDInWfOHGEHNBuCfJ+2EhO3XbN3Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"07007ea7e436a9727748dab75d8d1405967692aa0d95d61c649f1ea8ea565477","last_reissued_at":"2026-07-04T14:56:09.084607Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:56:09.084607Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The switching element for a Leonard pair","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RA","authors_text":"Kazumasa Nomura, Paul Terwilliger","submitted_at":"2006-08-25T00:27:39Z","abstract_excerpt":"Let $V$ denote a vector space with finite positive dimension. We consider a pair of linear transformations $A : V \\to V$ and $A^* : V \\to V$ 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 {\\em Leonard pair} on $V$. 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