{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nLet $F$ be a field and let $c_0, c_1, \\dots, c_{n-1} \\in F$.\nThe *circulant matrix* with first column $(c_0, c_1, \\dots, c_{n-1})^T$ is the $n \\times n$ matrix $C$ with entries $C_{ij} = c_{(i - j) \\bmod n}$, for $0 \\le i, j \\le n - 1$.\nEach column is the previous column shifted down by one place, with the last entry moved to the top.\nFor $n = 3$:\n$$C = \\begin{pmatrix} c_0 \u0026 c_2 \u0026 c_1 \\\\ c_1 \u0026 c_0 \u0026 c_2 \\\\ c_2 \u0026 c_1 \u0026 c_0 \\end{pmatrix}.$$\n\n## Remarks\n\nSome authors define circulant matrices by their first row; the two conventions give transposed matrices.\nOver $\\C$, the vectors $(1, \\omega^k, \\omega^{2k}, \\dots, \\omega^{(n-1)k})$ with $\\omega = e^{2\\pi i/n}$ are eigenvectors of every $n \\times n$ circulant matrix.\nThe product of two circulant matrices of the same order is circulant, and circulant matrices commute with each other.\n\n## Explanations\n\n### L1 human\n\nIn a circulant matrix, each column is the column before it, rotated down by one step.\nAll the information is in the first column.\nThe discrete Fourier transform diagonalises all of these matrices at the same time.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "instance_of": [
      "0JXZKX526R"
    ],
    "kind": "example",
    "msc": [
      "15B05"
    ],
    "name": "Circulant matrix",
    "same_as": {
      "wikidata": "Q245457",
      "wikipedia": "Circulant matrix"
    },
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Circulant_matrix"
      }
    ],
    "tag": "07HFRH2AWY",
    "uses": [
      "0CP7YY6NAG"
    ]
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/07HFRH2AWY/",
  "links": [
    {
      "field": "instance_of",
      "target": "0JXZKX526R"
    },
    {
      "field": "uses",
      "target": "0CP7YY6NAG"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/07HFRH2AWY.md",
  "msc": [
    "15B05"
  ],
  "name": "Circulant matrix",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/07H/07HFRH2AWY.md",
  "status": null,
  "tag": "07HFRH2AWY",
  "uri": "m2://07HFRH2AWY",
  "url": "https://math2.tamnd.com/wiki/circulant-matrix/",
  "writing": {
    "tag": "07HFRH2AWY",
    "kind": "example",
    "level": "W2",
    "next": [
      {
        "what": "a person confirmed each agent-written section",
        "note": "rule P04",
        "have": 0,
        "want": 1
      }
    ]
  }
}