{
  "id": "1407.2485",
  "version": "v1",
  "published": "2014-07-09T14:04:34.000Z",
  "updated": "2014-07-09T14:04:34.000Z",
  "title": "Strong Shift Equivalence and Positive Doubly Stochastic Matrices",
  "authors": [
    "Sompong Chuysurichay"
  ],
  "comment": "12 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We give sufficient conditions for a positive stochastic matrix to be similar and strong shift equivalent over $\\mathbb{R}_+$ to a positive doubly stochastic matrix through matrices of the same size. We also prove that every positive stochastic matrix is strong shift equivalent over $\\mathbb{R}_+$ to a positive doubly stochastic matrix. Consequently, the set of nonzero spectra of primitive stochastic matrices over $\\mathbb{R}$ with positive trace and the set of nonzero spectra of positive doubly stochastic matrices over $\\mathbb{R}$ are identical. We exhibit a class of $2\\times 2$ matrices, pairwise strong shift equivalent over $\\mathbb R_+$ through $2\\times 2$ matrices, for which there is no uniform upper bound on the minimum lag of a strong shift equivalence through matrices of bounded size. In contrast, we show for any $n\\times n$ primitive matrix of positive trace that the set of positive $n\\times n$ matrices similar to it contains only finitely many SSE-$\\mathbb R_+$ classes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-09T14:04:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B48",
      "37B10",
      "15A21",
      "15B51"
    ],
    "keywords": [
      "positive doubly stochastic matrices",
      "strong shift equivalence",
      "strong shift equivalent",
      "positive doubly stochastic matrix",
      "positive stochastic matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.2485C"
    }
  }
}