{
  "id": "1907.03368",
  "version": "v1",
  "published": "2019-07-07T23:47:26.000Z",
  "updated": "2019-07-07T23:47:26.000Z",
  "title": "Minimal curves in U(n) and Gl(n)+ with respect to the spectral and the trace norms",
  "authors": [
    "Jorge Antezana",
    "Eduardo Ghiglioni",
    "Demetrio Stojanoff"
  ],
  "categories": [
    "math.FA",
    "math.AG",
    "math.MG"
  ],
  "abstract": "Consider the Lie group of n x n complex unitary matrices U(n) endowed with the bi-invariant Finsler metric given by the spectral norm, ||X||_U = ||U*X||_{sp} = ||X||_{sp} for any X tangent to a unitary operator U. Given two points in U(n), in general there exists infinitely many curves of minimal length. The aim of this paper is to provide a complete description of such curves. As a consequence of this description, we conclude that there is a unique curve of minimal length between U and V if and only if the spectrum of U*V is contained in a set of the form \\{e^{i \\theta}, e^{-i \\theta}\\} for some \\theta \\in [0, \\infty). Similar studies are done for the Grassmann manifolds. Now consider the cone of n x n positive invertible matrices Gl(n)+ endowed with the bi-invariant Finsler metric given by the trace norm, ||X||_{1, A} = ||A^{-1/2}XA^{-1/2}||_1 for any X tangent to A \\in Gl(n)+. In this context, given two points A,B \\in Gl(n)+ there exists infinitely many curves of minimal length. In order to provide a complete description of such curves, we provide a characterization of the minimal curves joining two Hermitian matrices X, Y \\in H(n). As a consequence of the last description, we provide a way to construct minimal paths in the group of unitary matrices U(n) endowed with the bi-invariant Finsler metric ||X||_{1, U} = ||U*X||_{1} = ||X||_{1} for any X tangent to U \\in U(n). We also study the set of intermediate points in all the previous contexts. Between two given unitary matrices U and V we prove that this set is geodesically convex provided ||U - V||_{sp} < 1. In Gl(n)+ this set is geodesically convex for every unitarily invariant norm.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-07T23:47:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "trace norm",
      "minimal curves",
      "bi-invariant finsler metric",
      "minimal length",
      "complete description"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}