{
  "id": "0903.0614",
  "version": "v1",
  "published": "2009-03-03T20:46:21.000Z",
  "updated": "2009-03-03T20:46:21.000Z",
  "title": "Random matrices: The distribution of the smallest singular values",
  "authors": [
    "Terence Tao",
    "Van Vu"
  ],
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Let $\\a$ be a real-valued random variable of mean zero and variance 1. Let $M_n(\\a)$ denote the $n \\times n$ random matrix whose entries are iid copies of $\\a$ and $\\sigma_n(M_n(\\a))$ denote the least singular value of $M_n(\\a)$. ($\\sigma_n(M_n(\\a))^2$ is also usually interpreted as the least eigenvalue of the Wishart matrix $M_n M_n^{\\ast}$.) We show that (under a finite moment assumption) the probability distribution $n \\sigma_n(M_n(\\a))^2$ is {\\it universal} in the sense that it does not depend on the distribution of $\\a$. In particular, it converges to the same limiting distribution as in the special case when $a$ is real gaussian. (The limiting distribution was computed explicitly in this case by Edelman.) We also proved a similar result for complex-valued random variables of mean zero, with real and imaginary parts having variance 1/2 and covariance zero. Similar results are also obtained for the joint distribution of the bottom $k$ singular values of $M_n(\\a)$ for any fixed $k$ (or even for $k$ growing as a small power of $n$) and for rectangular matrices. Our approach is motivated by the general idea of \"property testing\" from combinatorics and theoretical computer science. This seems to be a new approach in the study of spectra of random matrices and combines tools from various areas of mathematics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-03-03T20:46:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "smallest singular values",
      "random matrices",
      "mean zero",
      "similar result",
      "limiting distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.0614T"
    }
  }
}