{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "JJT6GC055J"
    },
    {
      "field": "defines",
      "target": "K33BQX70N0"
    }
  ],
  "body": "\n## Summary\n\nThe Frobenius norm of a matrix is the square root of the sum of the squares of the absolute values of its entries.\nIt is the Euclidean length of the matrix seen as a long vector, and it comes from the inner product $\\ip{A}{B} = \\tr(B^* A)$.\nIt is a submultiplicative [[matrix norm|54T6Y6DSY4]] that is easy to compute.\nIt equals the square root of the sum of the squares of the [[singular values|09VMDSRXWJ]].\n\n## Explanations\n\n### L1 human\n\nWrite all entries of a matrix in one long row and measure the length of that row as in the theorem of Pythagoras.\nThat length is the Frobenius norm.\nFor $\\begin{pmatrix} 1 \u0026 2 \\\\ 3 \u0026 4 \\end{pmatrix}$ it is $\\sqrt{1 + 4 + 9 + 16} = \\sqrt{30}$.\nIt is used to measure how far one matrix is from another, for example in least squares fitting and low rank approximation.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "15A60"
    ],
    "name": "Frobenius norm",
    "sources": [
      {
        "locator": "Frobenius norm",
        "url": "https://en.wikipedia.org/wiki/Matrix_norm"
      }
    ],
    "specializes": [
      "54T6Y6DSY4"
    ],
    "tag": "JT8Q204E4R"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/JT8Q204E4R/",
  "links": [
    {
      "field": "specializes",
      "target": "54T6Y6DSY4"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/JT8Q204E4R.md",
  "msc": [
    "15A60"
  ],
  "name": "Frobenius norm",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JT8/JT8Q204E4R.md",
  "status": null,
  "tag": "JT8Q204E4R",
  "uri": "m2://JT8Q204E4R",
  "url": "https://math2.tamnd.com/wiki/frobenius-norm/",
  "writing": {
    "tag": "JT8Q204E4R",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}