{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0WFED3K3DQ"
    }
  ],
  "body": "\n## Statement\n\nLet $F$ be a field and let $V$ be a finite-dimensional vector space over $F$, with two bases $\\mathcal{B}$ and $\\mathcal{B}'$.\nLet $P$ be the [[change of basis matrix|0MKRHZB3WS]] whose columns are the coordinate vectors of the vectors of $\\mathcal{B}'$ in $\\mathcal{B}$.\nLet $B$ be a bilinear form on $V$, with [[matrix|0QK6K857HS]] $M$ in $\\mathcal{B}$ and $M'$ in $\\mathcal{B}'$.\nThen\n$$M' = P^T M P.$$\n\n## Remarks\n\nTwo square matrices $M$ and $M'$ are *congruent* when $M' = P^T M P$ for an invertible matrix $P$.\nSo two matrices represent the same bilinear form in two bases if and only if they are congruent.\nCongruence keeps the rank and the symmetry of a matrix, but in general not its eigenvalues.\nCompare this with the rule $P^{-1} A P$ for the matrix of a linear map.\n\n## Explanations\n\n### L1 human\n\nWhen we change the basis, the matrix of a bilinear form changes by $P^T M P$.\nBoth vectors change coordinates, so $P$ appears twice.\nFor a linear map, the rule is $P^{-1} A P$ instead.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "15A63",
      "15A21"
    ],
    "name": "Change of basis for a bilinear form",
    "role": "proposition",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Matrix_congruence"
      }
    ],
    "tag": "0SPYAGE7T0",
    "uses": [
      "0P24YW565F",
      "0KWXYZPJKN"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0SPYAGE7T0/",
  "links": [
    {
      "field": "uses",
      "target": "0P24YW565F"
    },
    {
      "field": "uses",
      "target": "0KWXYZPJKN"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0SPYAGE7T0.md",
  "msc": [
    "15A63",
    "15A21"
  ],
  "name": "Change of basis for a bilinear form",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0SP/0SPYAGE7T0.md",
  "status": null,
  "tag": "0SPYAGE7T0",
  "uri": "m2://0SPYAGE7T0",
  "url": "https://math2.tamnd.com/wiki/change-of-basis-for-a-bilinear-form/",
  "writing": {
    "tag": "0SPYAGE7T0",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}