{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "011K11VPJN"
    }
  ],
  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $T \\colon V \\to V$ be an [[endomorphism|0J7H0YBFDJ]].\nThe map $T$ is *nilpotent* when $T^k = 0$ for some integer $k \\ge 1$.\nThe least such $k$ is the *index* of $T$.\n\n## Remarks\n\nIf $\\dim V = n$ is finite and $T$ is nilpotent, then $T^n = 0$.\nIn finite dimensions, $T$ is nilpotent if and only if its characteristic polynomial is $t^n$.\nOver an algebraically closed field, this means that $0$ is its only eigenvalue.\nA matrix is nilpotent when its map is nilpotent, see [[nilpotent matrix|07BSPZPSMF]].\nThe derivative on the space of polynomials of degree less than $n$ is nilpotent.\n\n## Explanations\n\n### L1 human\n\nA nilpotent map destroys everything after a few steps.\nEach application pushes vectors further towards zero, until they vanish.\nThe shift $e_1 \\mapsto 0$, $e_2 \\mapsto e_1$, $e_3 \\mapsto e_2$ in $F^3$ is nilpotent of index $3$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "063D5Z2C70",
    "kind": "definition",
    "msc": [
      "15A04",
      "15A21"
    ],
    "name": "Nilpotent endomorphism",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Nilpotent_matrix"
      }
    ],
    "tag": "0M8MHYERTY",
    "uses": [
      "0PTF1PTJK0"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0M8MHYERTY/",
  "links": [
    {
      "field": "defines",
      "target": "063D5Z2C70"
    },
    {
      "field": "uses",
      "target": "0PTF1PTJK0"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0M8MHYERTY.md",
  "msc": [
    "15A04",
    "15A21"
  ],
  "name": "Nilpotent endomorphism",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0M8/0M8MHYERTY.md",
  "status": null,
  "tag": "0M8MHYERTY",
  "uri": "m2://0M8MHYERTY",
  "url": "https://math2.tamnd.com/wiki/nilpotent-endomorphism-definition/",
  "writing": {
    "tag": "0M8MHYERTY",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}