{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \\colon V \\to V$ be a linear map with [[characteristic polynomial|0J5EMJRRFM]] $p_T$.\nThen $p_T(T) = 0$, where $p_T(T)$ is the [[value of $p_T$ at $T$|0E3MV6E6MG]].\n\n## Remarks\n\nIt follows from the [[Cayley–Hamilton theorem for matrices|0F765KFC2J]] applied to the matrix of $T$ in a basis.\nThe map $p \\mapsto p([T]_B)$ corresponds to $p \\mapsto p(T)$ under $T \\mapsto [T]_B$.\nThe statement also holds for endomorphisms of finitely generated modules over a commutative ring, with a suitable characteristic polynomial.\n\n## Explanations\n\n### L1 human\n\nThe theorem does not depend on a basis.\nA linear map satisfies its characteristic polynomial, because each of its matrices does.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "kind": "statement",
    "msc": [
      "15A04",
      "15A18"
    ],
    "name": "Cayley–Hamilton theorem for linear maps",
    "role": "theorem",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Cayley–Hamilton_theorem"
      }
    ],
    "tag": "0QEM68Y0DE",
    "uses": [
      "0BZ81AJNWF",
      "0BNEXSKNRV"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0QEM68Y0DE/",
  "links": [
    {
      "field": "uses",
      "target": "0BZ81AJNWF"
    },
    {
      "field": "uses",
      "target": "0BNEXSKNRV"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0QEM68Y0DE.md",
  "msc": [
    "15A04",
    "15A18"
  ],
  "name": "Cayley–Hamilton theorem for linear maps",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0QE/0QEM68Y0DE.md",
  "status": null,
  "tag": "0QEM68Y0DE",
  "uri": "m2://0QEM68Y0DE",
  "url": "https://math2.tamnd.com/wiki/cayley-hamilton-theorem-for-linear-maps/",
  "writing": {
    "tag": "0QEM68Y0DE",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}