{
  "id": "1804.08140",
  "version": "v1",
  "published": "2018-04-22T17:23:22.000Z",
  "updated": "2018-04-22T17:23:22.000Z",
  "title": "Spectral properties of Kac-Murdock-Szegö matrices with a complex parameter",
  "authors": [
    "George Fikioris"
  ],
  "comment": "1 figure",
  "categories": [
    "math.NA"
  ],
  "abstract": "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 and pinpoint all eigenvalues' locations. Our study starts from a certain polynomial whose zeros are connected to the eigenvalues by elementary formulas. We discuss relations of our results to earlier results of W. F. Trench.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-22T17:23:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B05",
      "15A18",
      "65F15"
    ],
    "keywords": [
      "spectral properties",
      "complex parameter",
      "eigenvalues",
      "toeplitz correlation matrix",
      "simple approximate formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}