{
  "id": "2210.15320",
  "version": "v1",
  "published": "2022-10-27T11:00:35.000Z",
  "updated": "2022-10-27T11:00:35.000Z",
  "title": "On Hadamard powers of Random Wishart matrices",
  "authors": [
    "Jnaneshwar Baslingker"
  ],
  "comment": "Comments are welcome",
  "categories": [
    "math.PR"
  ],
  "abstract": "A famous result of Horn and Fitzgerald is that the $\\beta$-th Hadamard power of any $n\\times n$ positive semi-definite (p.s.d) matrix with non-negative entries is p.s.d $\\forall \\beta\\geq n-2$ and is not necessarliy p.s.d for $\\beta< n-2,$ with $\\ \\beta\\notin \\mathbb{N}$. In this article, we study this question for random Wishart matrix $A_n:={X_nX_n^T}$, where $X_n$ is $n\\times n$ matrix with i.i.d. Gaussians. It is shown that applying $x\\rightarrow |x|^{\\alpha}$ entrywise to $A_n$, the resulting matrix is p.s.d, with high probability, for $\\alpha>1$ and is not p.s.d, with high probability, for $\\alpha<1$. It is also shown that if $X_n$ are $\\lfloor n^{s}\\rfloor\\times n$ matrices, for any $s<1$, the transition of positivity occurs at the exponent $\\alpha=s$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-27T11:00:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "60B11"
    ],
    "keywords": [
      "random wishart matrix",
      "high probability",
      "th hadamard power",
      "positivity occurs",
      "non-negative entries"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}