{
  "id": "1312.6048",
  "version": "v1",
  "published": "2013-12-20T17:10:19.000Z",
  "updated": "2013-12-20T17:10:19.000Z",
  "title": "Minimum ranks of sign patterns via sign vectors and duality",
  "authors": [
    "Marina Arav",
    "Frank J. Hall",
    "Zhongshan Li",
    "Hein van der Holst",
    "John Sinkovic",
    "Lihua Zhang"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A {\\it sign pattern matrix} is a matrix whose entries are from the set $\\{+,-, 0\\}$. The minimum rank of a sign pattern matrix $A$ is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of $A$. It is shown in this paper that for any $m \\times n$ sign pattern $A$ with minimum rank $n-2$, rational realization of the minimum rank is possible. This is done using a new approach involving sign vectors and duality. It is shown that for each integer $n\\geq 9$, there exists a nonnegative integer $m$ such that there exists an $n\\times m$ sign pattern matrix with minimum rank $n-3$ for which rational realization is not possible. A characterization of $m\\times n$ sign patterns $A$ with minimum rank $n-1$ is given (which solves an open problem in Brualdi et al. \\cite{Bru10}), along with a more general description of sign patterns with minimum rank $r$, in terms of sign vectors of certain subspaces. A number of results on the maximum and minimum numbers of sign vectors of $k$-dimensional subspaces of $\\mathbb R^n$ are obtained. In particular, it is shown that the maximum number of sign vectors of $2$-dimensional subspaces of $\\mathbb R^n$ is $4n+1$. Several related open problems are stated along the way.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-20T17:10:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B35",
      "15B36",
      "52C40"
    ],
    "keywords": [
      "minimum rank",
      "sign vectors",
      "sign pattern matrix",
      "dimensional subspaces",
      "rational realization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.6048A"
    }
  }
}