{"paper":{"title":"Double, borderline, and extraordinary eigenvalues of Kac-Murdock-Szeg\\\"o matrices with a complex parameter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.NA","authors_text":"George Fikioris, Themistoklis K. Mavrogordatos","submitted_at":"2018-12-16T10:25:32Z","abstract_excerpt":"For all sufficiently large complex $\\rho$, and for arbitrary matrix dimension $n$, it is shown that the Kac--Murdock--Szeg\\H{o} matrix $K_n(\\rho)=\\left[\\rho^{|j-k|}\\right]_{j,k=1}^{n}$ possesses exactly two eigenvalues whose magnitude is larger than $n$. We discuss a number of properties of the two \"extraordinary\" eigenvalues. Conditions are developed that, given $n$, allow us-without actually computing eigenvalues-to find all values $\\rho$ that give rise to eigenvalues of magnitude $n$, termed \"borderline\" eigenvalues. The aforementioned values of $\\rho$ form two closed curves in the complex-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.06437","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"}