{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "C37R11WREF"
    },
    {
      "field": "uses",
      "target": "NKTAN9XBZA"
    },
    {
      "field": "uses",
      "target": "V2H578D4RT"
    },
    {
      "field": "defines",
      "target": "XDQ8CF0EZD"
    }
  ],
  "body": "\n## Summary\n\nEvery [[positive operator|77020CY5ZW]] $A$ on a Hilbert space has exactly one positive operator $B$ with $B^2 = A$, written $A^{1/2}$ or $\\sqrt{A}$.\nIt extends the [[square root of a positive semidefinite matrix|0BJVDCF38Y]] to infinite dimensions.\nThe square root defines the absolute value $\\abs{T} = (T^*T)^{1/2}$ of an operator, which enters the polar decomposition and the definition of singular values.\n\n## Explanations\n\n### L1 human\n\nA positive operator behaves like a nonnegative number, so it should have a square root that is again nonnegative.\nFor a diagonal operator with nonnegative entries $d_1, d_2, \\ldots$, the square root is the diagonal operator with entries $\\sqrt{d_1}, \\sqrt{d_2}, \\ldots$.\nAn operator can have many square roots, for instance $I$ and $-I$ both square to $I$, but only one of them is positive.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "47A60",
      "47B65"
    ],
    "name": "Square root of a positive operator",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Square_root_of_a_matrix"
      }
    ],
    "tag": "3JY5HNFRMF"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/3JY5HNFRMF/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/3JY5HNFRMF.md",
  "msc": [
    "47A60",
    "47B65"
  ],
  "name": "Square root of a positive operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3JY/3JY5HNFRMF.md",
  "status": null,
  "tag": "3JY5HNFRMF",
  "uri": "m2://3JY5HNFRMF",
  "url": "https://math2.tamnd.com/wiki/square-root-of-a-positive-operator/",
  "writing": {
    "tag": "3JY5HNFRMF",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}