{
  "id": "1805.06999",
  "version": "v1",
  "published": "2018-05-18T00:25:17.000Z",
  "updated": "2018-05-18T00:25:17.000Z",
  "title": "On the singular value decomposition over finite fields and orbits of GU x GU",
  "authors": [
    "Robert M. Guralnick"
  ],
  "comment": "12 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GU(m,q) x GU(n,q) on n by n matrices (which is the analog of the singular value decomposition). The proof involves Kronecker's theory of pencils and the Lang-Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick, Larsen and Tiep where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form AB where B is either the transpose of A or the conjugate transpose.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-18T00:25:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B57",
      "15A18",
      "15A22",
      "20G15",
      "20G40"
    ],
    "keywords": [
      "singular value decomposition",
      "finite field",
      "fundamental concept",
      "linear algebra",
      "form ab"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}