{"paper":{"title":"Determinantal and eigenvalue inequalities for matrices with numerical ranges in a sector","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Chi-Kwong Li, Nung-sing Sze","submitted_at":"2013-06-20T15:41:00Z","abstract_excerpt":"Let $A = \\pmatrix A_{11} & A_{12} \\cr A_{21} & A_{22}\\cr\\pmatrix \\in M_n$, where $A_{11} \\in M_m$ with $m \\le n/2$, be such that the numerical range of $A$ lies in the set $\\{e^{i\\varphi} z \\in \\IC: |\\Im z| \\le (\\Re z) \\tan \\alpha\\}$, for some $\\varphi \\in [0, 2\\pi)$ and $\\alpha \\in [0, \\pi/2)$. We obtain the optimal containment region for the generalized eigenvalue $\\lambda$ satisfying $$\\lambda \\pmatrix A_{11} & 0 \\cr 0 & A_{22}\\cr\\pmatrix x = \\pmatrix 0 & A_{12} \\cr A_{21} & 0\\cr\\pmatrix x \\quad \\hbox{for some nonzero} x \\in \\IC^n,$$ and the optimal eigenvalue containment region of the matr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.4916","kind":"arxiv","version":3},"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"}