{
  "id": "1901.08655",
  "version": "v1",
  "published": "2019-01-24T21:53:39.000Z",
  "updated": "2019-01-24T21:53:39.000Z",
  "title": "Small ball probability for the condition number of random matrices",
  "authors": [
    "Alexander E. Litvak",
    "Konstantin Tikhomirov",
    "Nicole Tomczak-Jaegermann"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $A$ be an $n\\times n$ random matrix with i.i.d. entries of zero mean, unit variance and a bounded subgaussian moment. We show that the condition number $s_{\\max}(A)/s_{\\min}(A)$ satisfies the small ball probability estimate $${\\mathbb P}\\big\\{s_{\\max}(A)/s_{\\min}(A)\\leq n/t\\big\\}\\leq 2\\exp(-c t^2),\\quad t\\geq 1,$$ where $c>0$ may only depend on the subgaussian moment. Although the estimate can be obtained as a combination of known results and techniques, it was not noticed in the literature before. As a key step of the proof, we apply estimates for the singular values of $A$, ${\\mathbb P}\\big\\{s_{n-k+1}(A)\\leq ck/\\sqrt{n}\\big\\}\\leq 2 \\exp(-c k^2), \\quad 1\\leq k\\leq n,$ obtained (under some additional assumptions) by Nguyen.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-24T21:53:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "condition number",
      "random matrix",
      "small ball probability estimate",
      "unit variance",
      "additional assumptions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}