{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "SAQ5EFQP7H"
    }
  ],
  "body": "\n## Summary\n\nThe frame operator of a frame $(f_i)$ in a [[Hilbert space|HM2HHE3J51]] sends a vector $f$ to $\\sum_i \\ip{f}{f_i} f_i$.\nIt is bounded, [[self-adjoint|0NEGN5CRSC]], [[positive|77020CY5ZW]] and [[invertible|CPAD5W1CGW]], and its spectrum lies between the frame bounds.\nIts inverse gives the canonical dual frame, and so a formula that rebuilds every vector from its frame coefficients.\n\n## Explanations\n\n### L1 human\n\nA frame lets one measure a vector by taking inner products with many fixed vectors.\nThe frame operator first measures, then adds up the fixed vectors weighted by the measurements.\nFor an orthonormal basis this gives back the vector itself.\nFor a general frame the result is distorted, but in a way that can be undone, because the operator is invertible.\nFor example, the three Mercedes-Benz star vectors in the plane give the frame operator $\\tfrac32 I$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "42C15",
      "47B02"
    ],
    "name": "Frame operator",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Frame_(linear_algebra)"
      }
    ],
    "tag": "BP3WR8NG1W"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/BP3WR8NG1W/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/BP3WR8NG1W.md",
  "msc": [
    "42C15",
    "47B02"
  ],
  "name": "Frame operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BP3/BP3WR8NG1W.md",
  "status": null,
  "tag": "BP3WR8NG1W",
  "uri": "m2://BP3WR8NG1W",
  "url": "https://tamnd.github.io/math2/wiki/frame-operator/",
  "writing": {
    "tag": "BP3WR8NG1W",
    "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
      }
    ]
  }
}