{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0Y95Z7MT02"
    }
  ],
  "body": "\n## Statement\n\nLet $F$ be a field and let $A$ be an $n \\times n$ matrix over $F$.\nThen $A$ is [[diagonalisable|07SQCNV233]] if and only if $F^n$ has a [[basis|0KCQ9Y3QKA]] of [[eigenvectors|0N6960W0BV]] of $A$.\nIf $v_1, \\dots, v_n$ is such a basis with $A v_i = \\lambda_i v_i$, and $P$ has the columns $v_1, \\dots, v_n$, then $P^{-1} A P$ is the diagonal matrix with entries $\\lambda_1, \\dots, \\lambda_n$.\n\n## Remarks\n\nThe equation $AP = PD$ says column by column that $A v_i = \\lambda_i v_i$.\nEquivalently, $F^n$ is the direct sum of the eigenspaces of $A$.\nEquivalently, the sum of the [[geometric multiplicities|099TWVJ909]] of the eigenvalues is $n$.\n\n## Explanations\n\n### L1 human\n\nTo diagonalise a matrix, we need $n$ independent eigenvectors.\nWe put them as columns into a matrix $P$.\nThen $P^{-1} A P$ is diagonal, with the eigenvalues on the diagonal.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "15A20"
    ],
    "name": "A matrix is diagonalisable iff it has a basis of eigenvectors",
    "role": "theorem",
    "sources": [
      {
        "locator": "Characterization",
        "url": "https://en.wikipedia.org/wiki/Diagonalizable_matrix"
      }
    ],
    "tag": "056SW1E0AC",
    "uses": [
      "0Y17PEKF24",
      "0B5H4208Z9",
      "0VTQTPQGAV"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/056SW1E0AC/",
  "links": [
    {
      "field": "uses",
      "target": "0Y17PEKF24"
    },
    {
      "field": "uses",
      "target": "0B5H4208Z9"
    },
    {
      "field": "uses",
      "target": "0VTQTPQGAV"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/056SW1E0AC.md",
  "msc": [
    "15A20"
  ],
  "name": "A matrix is diagonalisable iff it has a basis of eigenvectors",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/056/056SW1E0AC.md",
  "status": null,
  "tag": "056SW1E0AC",
  "uri": "m2://056SW1E0AC",
  "url": "https://math2.tamnd.com/wiki/a-matrix-is-diagonalisable-iff-it-has-a-basis-of-eigenvectors/",
  "writing": {
    "tag": "056SW1E0AC",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}