{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "011K11VPJN"
    }
  ],
  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \\colon V \\to V$ be an endomorphism.\nThe map $T$ is *semisimple* when, for each $T$-[[invariant subspace|0ACX5QT8TE]] $W$ of $V$, there is a $T$-invariant subspace $W'$ with $V = W \\oplus W'$.\n\n## Remarks\n\nThe map $T$ is semisimple if and only if its minimal polynomial has no repeated irreducible factor.\nOver an algebraically closed field, $T$ is semisimple if and only if it is [[diagonalisable|07SQCNV233]].\nThe rotation of $\\R^2$ by $90$ degrees is semisimple, but it is not diagonalisable over $\\R$.\n\n## Explanations\n\n### L1 human\n\nSemisimple maps generalise diagonalisable maps to fields where eigenvalues can be missing.\nThey have no hidden nilpotent part.\nOver $\\C$, semisimple simply means diagonalisable.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "08VRX3Q4PB",
    "kind": "definition",
    "msc": [
      "15A04",
      "15A20"
    ],
    "name": "Semisimple endomorphism",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Semisimple_operator"
      }
    ],
    "tag": "0QS8P74F2Z",
    "uses": [
      "0EP59ESQ94"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0QS8P74F2Z/",
  "links": [
    {
      "field": "defines",
      "target": "08VRX3Q4PB"
    },
    {
      "field": "uses",
      "target": "0EP59ESQ94"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0QS8P74F2Z.md",
  "msc": [
    "15A04",
    "15A20"
  ],
  "name": "Semisimple endomorphism",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0QS/0QS8P74F2Z.md",
  "status": null,
  "tag": "0QS8P74F2Z",
  "uri": "m2://0QS8P74F2Z",
  "url": "https://math2.tamnd.com/wiki/semisimple-endomorphism-definition/",
  "writing": {
    "tag": "0QS8P74F2Z",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}