{
  "id": "1606.05184",
  "version": "v1",
  "published": "2016-06-16T13:47:16.000Z",
  "updated": "2016-06-16T13:47:16.000Z",
  "title": "A Complete Characterization of Quadratic Polynomials that are determinants of Linear Matrix Polynomials",
  "authors": [
    "Papri Dey",
    "Harish K. Pillai"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "The problem of expressing a multivariate polynomial as the determinant of a monic (definite) symmetric (Hermitian) linear matrix polynomial (LMP) has drawn a huge amount of attention due to its connection with optimization problems. In this paper we provide a necessary and sufficient condition for the existence of monic symmetric determinantal representation (MSDR) of size 2 for a given quadratic polynomial. Further if an MSDR exists we propose a method to construct such a $2 \\times 2$ MSDR. It is known that a quadratic polynomial $f(x) = x^TAx + b^T x + 1$ has an MSDR of size $n + 1$ if $A$ is negative semidefinite. We prove that if a quadratic polynomial f(x) with A which is not negative semidefinite has an MSDR of size greater than 2, then it has an MSDR of size 2 too. We also characterize quadratic polynomials which exhibit diagonal MSDRs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-16T13:47:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quadratic polynomial",
      "linear matrix polynomial",
      "complete characterization",
      "monic symmetric determinantal representation",
      "negative semidefinite"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}