{"paper":{"title":"Maximum of the resolvent over matrices with given spectrum","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.SP"],"primary_cat":"math.NA","authors_text":"Oleg Szehr, Rachid Zarouf","submitted_at":"2015-01-28T07:23:40Z","abstract_excerpt":"In numerical analysis it is often necessary to estimate the condition number $CN(T)=||T||_{} \\cdot||T^{-1}||_{}$ and the norm of the resolvent $||(\\zeta-T)^{-1}||_{}$ of a given $n\\times n$ matrix $T$. We derive new spectral estimates for these quantities and compute explicit matrices that achieve our bounds. We recover the well-known fact that the supremum of $CN(T)$ over all matrices with $||T||_{} \\leq1$ and minimal absolute eigenvalue $r=\\min_{i=1,...,n}|\\lambda_{i}|>0$ is the Kronecker bound $\\frac{1}{r^{n}}$. This result is subsequently generalized by computing the corresponding supremum"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.07007","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"}