{"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. Denton, Stephen J. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.03795","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.03795/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}