{
  "id": "math/0204312",
  "version": "v2",
  "published": "2002-04-25T12:58:35.000Z",
  "updated": "2004-02-21T23:45:17.000Z",
  "title": "On the universality of the probability distribution of the product $B^{-1}X$ of random matrices",
  "authors": [
    "Joshua Feinberg"
  ],
  "comment": "latex, 15 pages, added a remark concerning the relation of this work to matrix variate t-distributions, added references",
  "categories": [
    "math.PR",
    "cond-mat.dis-nn",
    "math-ph",
    "math.MP",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Consider random matrices $A$, of dimension $m\\times (m+n)$, drawn from an ensemble with probability density $f(\\rmtr AA^\\dagger)$, with $f(x)$ a given appropriate function. Break $A = (B,X)$ into an $m\\times m$ block $B$ and the complementary $m\\times n$ block $X$, and define the random matrix $Z=B^{-1}X$. We calculate the probability density function $P(Z)$ of the random matrix $Z$ and find that it is a universal function, independent of $f(x)$. The universal probability distribution $P(Z)$ is a spherically symmetric matrix-variate $t$-distribution. Universality of $P(Z)$ is, essentially, a consequence of rotational invariance of the probability ensembles we study. As an application, we study the distribution of solutions of systems of linear equations with random coefficients, and extend a classic result due to Girko.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-02-21T23:45:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A52",
      "60E05",
      "62H10",
      "34F05"
    ],
    "keywords": [
      "random matrices",
      "universality",
      "random matrix",
      "probability density function",
      "universal probability distribution"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......4312F"
    }
  }
}