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Let $A$ denote a matrix in $\\matR$ and let $\\{\\th_i\\}_{i=0}^d$ denote the roots of the characteristic polynomial of $A$. We say $A$ is multiplicity-free whenever these roots are mutually distinct and contained in $\\R$. In this case $E_i$ will denote the primitive idempotent of $A$ associated with $\\th_i$ $(0 \\leq i \\leq d)$. We say $A$ is symmetrizable whenever"},"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":"1010.1305","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-07T00:20:44Z","cross_cats_sorted":[],"title_canon_sha256":"78a1f794aa8d4b7f9baea23b9b1ca6636bf9482a32b9434b17d1a5de310de8c8","abstract_canon_sha256":"c728f61777c05679af12be534fd5f602b287a98ef6a91e056ec5bada836c3f1f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:41.453218Z","signature_b64":"hyoOpzI4YaPlOr9LN0sJPfDCnd59sjs2L/Or0dg8GQ0TxZOMLGJeKskNunZjFncfRddW5mxJXoT+aLCpnBgaBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cf0580d3f48f243c98b7d0567c72c76cd64aee1ff53073e711dcc3e5bbde63b8","last_reissued_at":"2026-05-18T04:39:41.452755Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:41.452755Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tridiagonal matrices with nonnegative entries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kazumasa Nomura, Paul Terwilliger","submitted_at":"2010-10-07T00:20:44Z","abstract_excerpt":"In this paper we characterize the nonnegative irreducible tridiagonal matrices and their permutations, using certain entries in their primitive idempotents. Our main result is summarized as follows. Let $d$ denote a nonnegative integer. Let $A$ denote a matrix in $\\matR$ and let $\\{\\th_i\\}_{i=0}^d$ denote the roots of the characteristic polynomial of $A$. We say $A$ is multiplicity-free whenever these roots are mutually distinct and contained in $\\R$. In this case $E_i$ will denote the primitive idempotent of $A$ associated with $\\th_i$ $(0 \\leq i \\leq d)$. 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