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We consider an ordered pair of linear transformations $A: V \\to V$ and $A^*: V \\to V$ which satisfy both (i), (ii) below. \\begin{enumerate} \\item There exists a basis for $V$ with respect to which the matrix representing $A$ is Hessenberg and the matrix representing $A^*$ is diagonal.\n  \\item There exists a basis for $V$ with respect to which the matrix representing $A$ is diagona"},"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":"0911.4118","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2009-11-20T20:54:07Z","cross_cats_sorted":[],"title_canon_sha256":"cb20d6eba2352826d8996877bb8269596c36a8adfb857119bb6a73933ca325b7","abstract_canon_sha256":"43d66c60755cba306e0baa2e02ef4b54e39cf32d876ce81c413490c467fd9aa9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T17:16:26.758635Z","signature_b64":"kXWgvXEt7QuabRp8KNKx5hgdZRH19yAQ6fkwA33R+fdWCLCPVd1dJopPWrgNFCqCNZ5z0omfdyolfQnYsmOEAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2d02b28b16e4911a3bed95ade46206971731d20721471d134b57f1eb47619f21","last_reissued_at":"2026-07-04T17:16:26.758210Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T17:16:26.758210Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Thin Hessenberg Pairs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Ali Godjali","submitted_at":"2009-11-20T20:54:07Z","abstract_excerpt":"A square matrix is called {\\it Hessenberg} whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. 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