{
  "id": "2305.15107",
  "version": "v1",
  "published": "2023-05-24T12:53:28.000Z",
  "updated": "2023-05-24T12:53:28.000Z",
  "title": "On the eigenvalues of Toeplitz matrices with two off-diagonals",
  "authors": [
    "Sven-Erik Ekström",
    "David Meadon"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "Consider the Toeplitz matrix $T_n(f)$ generated by the symbol $f(\\theta)=\\hat{f}_r e^{\\mathbf{i}r\\theta}+\\hat{f}_0+\\hat{f}_{-s} e^{-\\mathbf{i}s\\theta}$, where $\\hat{f}_r, \\hat{f}_0, \\hat{f}_{-s} \\in \\mathbb{C}$ and $0<r<n,~0<s<n$. For $r=s=1$ we have the classical tridiagonal Toeplitz matrices, for which the eigenvalues and eigenvectors are known. Similarly, the eigendecompositions are known for $1<r=s$, when the generated matrices are ``symmetrically sparse tridiagonal''. In the current paper we study the eigenvalues of $T_n(f)$ for $1\\leq r<s$, which are ``non-symmetrically sparse tridiagonal''. We propose an algorithm which constructs one or two ad hoc matrices smaller than $T_n(f)$, whose eigenvalues are sufficient for determining the full spectrum of $T_n(f)$. The algorithm is explained through use of a conjecture for which examples and numerical experiments are reported for supporting it and for clarifying the presentation. Open problems are briefly discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-24T12:53:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A18",
      "15B05",
      "65F15"
    ],
    "keywords": [
      "toeplitz matrix",
      "eigenvalues",
      "off-diagonals",
      "ad hoc matrices smaller",
      "classical tridiagonal toeplitz matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}