{"paper":{"title":"Bounds for the Rayleigh quotient and the spectrum of self-adjoint operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.SP"],"primary_cat":"math.NA","authors_text":"Andrew V. Knyazev, Merico E. Argentati, Peizhen Zhu","submitted_at":"2012-07-13T13:32:32Z","abstract_excerpt":"The absolute change in the Rayleigh quotient (RQ) is bounded in this paper in terms of the norm of the residual and the change in the vector. If $x$ is an eigenvector of a self-adjoint bounded operator $A$ in a Hilbert space, then the RQ of the vector $x$, denoted by $\\rho(x)$, is an exact eigenvalue of $A$. In this case, the absolute change of the RQ $|\\rho(x)-\\rho(y)|$ becomes the absolute error in an eigenvalue $\\rho(x)$ of $A$ approximated by the RQ $\\rho(y)$ on a given vector $y.$ There are three traditional kinds of bounds of the eigenvalue error: a priori bounds via the angle between ve"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3240","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}