{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "06YSJVR356"
    },
    {
      "field": "uses",
      "target": "0JHGAV9H25"
    }
  ],
  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a vector space over $F$, let $T \\colon V \\to V$ be an endomorphism, and let $\\lambda \\in F$.\nThe *eigenspace* of $T$ for $\\lambda$ is\n$$E_\\lambda = \\ker(T - \\lambda \\id_V) = \\{v \\in V : T(v) = \\lambda v\\}.$$\n\n## Remarks\n\nThe eigenspace is a subspace of $V$; it is the [[kernel|0S23E6YE83]] of a linear map.\nIts nonzero elements are the [[eigenvectors|0N6960W0BV]] of $T$ for $\\lambda$.\n$E_\\lambda \\ne 0$ if and only if $\\lambda$ is an eigenvalue of $T$.\nThe dimension of $E_\\lambda$ is the [[geometric multiplicity|099TWVJ909]] of $\\lambda$.\n\n## Explanations\n\n### L1 human\n\nThe eigenspace collects all the vectors that the map stretches by the same factor $\\lambda$, with the zero vector.\nThese vectors form a subspace, for example a line or a plane.\nOn this subspace, the map acts as multiplication by $\\lambda$.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "defines": "0G0RFPG8HR",
    "kind": "definition",
    "msc": [
      "15A18"
    ],
    "name": "Eigenspace",
    "sources": [
      {
        "locator": "Eigenspaces, geometric multiplicity, and the eigenbasis",
        "url": "https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors"
      }
    ],
    "tag": "0AYEPBQMAW",
    "uses": [
      "0B5H4208Z9",
      "02YFME3RGW"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0AYEPBQMAW/",
  "links": [
    {
      "field": "defines",
      "target": "0G0RFPG8HR"
    },
    {
      "field": "uses",
      "target": "0B5H4208Z9"
    },
    {
      "field": "uses",
      "target": "02YFME3RGW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0AYEPBQMAW.md",
  "msc": [
    "15A18"
  ],
  "name": "Eigenspace",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0AY/0AYEPBQMAW.md",
  "status": null,
  "tag": "0AYEPBQMAW",
  "uri": "m2://0AYEPBQMAW",
  "url": "https://math2.tamnd.com/wiki/eigenspace-definition/",
  "writing": {
    "tag": "0AYEPBQMAW",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}