{
  "id": "2004.06757",
  "version": "v1",
  "published": "2020-04-14T19:13:22.000Z",
  "updated": "2020-04-14T19:13:22.000Z",
  "title": "Least singular value and condition number of a square random matrix with i.i.d. rows",
  "authors": [
    "Matteo Gregoratti",
    "Davide Maran"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a square random matrix made by i.i.d. rows with any distribution and prove that, for any given dimension, the probability for the least singular value to be in [0; $\\epsilon$) is at least of order $\\epsilon$. This allows us to generalize a result about the expectation of the condition number that was proved in the case of centered gaussian i.i.d. entries: such an expectation is always infinite. Moreover, we get some additional results for some well-known random matrix ensembles, in particular for the isotropic log-concave case, which is proved to have the best behaving in terms of the well conditioning.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-14T19:13:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "square random matrix",
      "condition number",
      "singular value",
      "well-known random matrix ensembles",
      "isotropic log-concave case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}