{"paper":{"title":"Spectral properties of Kac-Murdock-Szeg\\\"o matrices with a complex parameter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"George Fikioris","submitted_at":"2018-04-22T17:23:22Z","abstract_excerpt":"When $0\\lt \\rho \\lt 1$, the Kac-Murdock-Szeg\\\"o matrix $K_n(\\rho)=\\left[\\rho^{\\lvert j-k \\rvert}\\right]_{j,k=1}^n$ is a Toeplitz correlation matrix with many applications and very well known spectral properties. We study the eigenvalues and eigenvectors of $K_n(\\rho)$ for the general case where $\\rho$ is complex, pointing out similarities and differences to the case $0\\lt \\rho \\lt 1$. We then specialize our results to real $\\rho$ with $\\rho \\gt 1$, emphasizing the continuity of the eigenvalues as functions of $\\rho$. For $\\rho \\gt 1$, we develop simple approximate formulas for the eigenvalues "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.08140","kind":"arxiv","version":1},"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"}