{
  "id": "2203.12579",
  "version": "v1",
  "published": "2022-03-23T17:41:18.000Z",
  "updated": "2022-03-23T17:41:18.000Z",
  "title": "Phase Factors in Singular Value Decomposition and Schmidt Decomposition",
  "authors": [
    "Chu Ryang Wie"
  ],
  "comment": "7 pages",
  "categories": [
    "math.NA",
    "cs.NA",
    "quant-ph"
  ],
  "abstract": "In singular value decomposition (SVD) of a complex matrix A, the singular vectors or the eigenvectors of AA{\\dag} and A{\\dag}A are unique up to complex phase factors. Thus, the two unitary matrices in SVD are unique up to diagonal matrices of phase factors, the phase-factor matrices. Also, the product of these two phase-factor matrices, or the product of phase factors of the corresponding singular vectors with the same singular value, is unique. In the Schmidt decomposition, a phase-factor matrix is a phase rotation operator acting on a subsystem alone. We summarize here three simple steps to consistently carry out the SVD and the Schmidt decomposition including the phase factors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-23T17:41:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "singular value decomposition",
      "schmidt decomposition",
      "phase-factor matrix",
      "complex phase factors",
      "phase rotation operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}