{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0Z3GDY89T5"
    }
  ],
  "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 *generalised eigenspace* of $T$ for $\\lambda$ is\n$$V_\\lambda = \\{v \\in V : (T - \\lambda \\id_V)^k (v) = 0 \\text{ for some } k \\ge 1\\}.$$\n\n## Remarks\n\nIt is a $T$-invariant subspace that contains the eigenspace $E_\\lambda$.\nIf $\\dim V = n$ is finite, then $V_\\lambda = \\ker (T - \\lambda \\id_V)^n$.\nIts dimension is the [[algebraic multiplicity|0FY42N02DV]] of $\\lambda$, in finite dimensions.\nIts nonzero elements are the [[generalised eigenvectors|01H6QKWGDH]] for $\\lambda$.\n\n## Explanations\n\n### L1 human\n\nThe generalised eigenspace is a larger version of the eigenspace.\nIt contains the vectors that become zero after we apply $T - \\lambda \\id$ several times.\nThese spaces always fill the space over $\\C$, even when the eigenspaces do not.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0AJ035ZTZE",
    "kind": "definition",
    "msc": [
      "15A18"
    ],
    "name": "Generalised eigenspace",
    "sources": [
      {
        "locator": "Overview and definition",
        "url": "https://en.wikipedia.org/wiki/Generalized_eigenvector"
      }
    ],
    "tag": "07PNDWA7CE",
    "uses": [
      "0SAAHC36J9"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/07PNDWA7CE/",
  "links": [
    {
      "field": "defines",
      "target": "0AJ035ZTZE"
    },
    {
      "field": "uses",
      "target": "0SAAHC36J9"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/07PNDWA7CE.md",
  "msc": [
    "15A18"
  ],
  "name": "Generalised eigenspace",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/07P/07PNDWA7CE.md",
  "status": null,
  "tag": "07PNDWA7CE",
  "uri": "m2://07PNDWA7CE",
  "url": "https://math2.tamnd.com/wiki/generalised-eigenspace-definition/",
  "writing": {
    "tag": "07PNDWA7CE",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}