{
  "id": "2109.11143",
  "version": "v1",
  "published": "2021-09-23T05:00:32.000Z",
  "updated": "2021-09-23T05:00:32.000Z",
  "title": "Recovering eigenvectors from the absolute value of their entries",
  "authors": [
    "Stefan Steinerberger",
    "Hau-Tieng Wu"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We consider the eigenvalue problem $Ax = \\lambda x$ where $A \\in \\mathbb{R}^{n \\times n}$ and the eigenvalue is also real $\\lambda \\in \\mathbb{R}$. If we are given $A$, $\\lambda$ and, additionally, the absolute value of the entries of $x$ (the vector $(|x_i|)_{i=1}^n$), is there a fast way to recover $x$? In particular, can this be done quicker than computing $x$ from scratch? This may be understood as a special case of the phase retrieval problem. We present a randomized algorithm which provably converges in expectation whenever $\\lambda$ is a simple eigenvalue. The problem should become easier when $|\\lambda|$ is large and we discuss another algorithm for that case as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-23T05:00:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "absolute value",
      "recovering eigenvectors",
      "phase retrieval problem",
      "fast way",
      "special case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}