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Parke, Terence Tao, Xining Zhang","submitted_at":"2019-08-10T18:24:44Z","abstract_excerpt":"If $A$ is an $n \\times n$ Hermitian matrix with eigenvalues $\\lambda_1(A),\\dots,\\lambda_n(A)$ and $i,j = 1,\\dots,n$, then the $j^{\\mathrm{th}}$ component $v_{i,j}$ of a unit eigenvector $v_i$ associated to the eigenvalue $\\lambda_i(A)$ is related to the eigenvalues $\\lambda_1(M_j),\\dots,\\lambda_{n-1}(M_j)$ of the minor $M_j$ of $A$ formed by removing the $j^{\\mathrm{th}}$ row and column by the formula $$ |v_{i,j}|^2\\prod_{k=1;k\\neq i}^{n}\\left(\\lambda_i(A)-\\lambda_k(A)\\right)=\\prod_{k=1}^{n-1}\\left(\\lambda_i(A)-\\lambda_k(M_j)\\right)\\,.$$ We refer to this identity as the \\emph{eigenvector-eigen"},"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":"1908.03795","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2019-08-10T18:24:44Z","cross_cats_sorted":["hep-ph","hep-th","math-ph","math.MP"],"title_canon_sha256":"4b67778c5fdfbeb920b93384070e68398f50de43ce29f5043ee1db9c7e546fd1","abstract_canon_sha256":"710947537c331b0944b995a70e34bd80292a4bc7f2a01be57a0c443cd86977d7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:18:03.465527Z","signature_b64":"E8iQXOcw9cD/NJWqKZfGCUCPIuQl7SSjThlwhAyav4ALSrBPwo/mS1CgHB7bBrCCxw+h/7tppqjonKIvB/I6Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b75bb8ee378f6844bc67799cf78a84e77450fc1f4f48744d27186a89022db151","last_reissued_at":"2026-07-05T02:18:03.465066Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:18:03.465066Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-ph","hep-th","math-ph","math.MP"],"primary_cat":"math.RA","authors_text":"Peter B. 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