{
  "id": "2312.06170",
  "version": "v1",
  "published": "2023-12-11T07:18:39.000Z",
  "updated": "2023-12-11T07:18:39.000Z",
  "title": "Spectral properties of flipped Toeplitz matrices",
  "authors": [
    "Giovanni Barbarino",
    "Sven-Erik Ekström",
    "Carlo Garoni",
    "David Meadon",
    "Stefano Serra-Capizzano",
    "Paris Vassalos"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We study the spectral properties of flipped Toeplitz matrices of the form $H_n(f)=Y_nT_n(f)$, where $T_n(f)$ is the $n\\times n$ Toeplitz matrix generated by the function $f$ and $Y_n$ is the $n\\times n$ exchange (or flip) matrix having $1$ on the main anti-diagonal and $0$ elsewhere. In particular, under suitable assumptions on $f$, we establish an alternating sign relationship between the eigenvalues of $H_n(f)$, the eigenvalues of $T_n(f)$, and the quasi-uniform samples of $f$. Moreover, after fine-tuning a few known theorems on Toeplitz matrices, we use them to provide localization results for the eigenvalues of $H_n(f)$. Our study is motivated by the convergence analysis of the minimal residual (MINRES) method for the solution of real non-symmetric Toeplitz linear systems of the form $T_n(f)\\mathbf x=\\mathbf b$ after pre-multiplication of both sides by $Y_n$, as suggested by Pestana and Wathen.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-11T07:18:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B05",
      "15A18",
      "65F10"
    ],
    "keywords": [
      "toeplitz matrix",
      "flipped toeplitz matrices",
      "spectral properties",
      "real non-symmetric toeplitz linear systems",
      "eigenvalues"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}