{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $F$ be a field and let $A$ be a matrix over $F$ in [[row echelon form|0VYCDGMJJJ]].\nA *pivot* of $A$ is the leading entry of a nonzero row of $A$, that is, the first nonzero entry of that row.\nA *pivot column* of $A$ is a column that contains a pivot.\nFor a general matrix $B$, take a row echelon form of $B$ that is row equivalent to $B$.\nThe pivot positions of $B$ are the positions of the pivots of this form.\n\n## Remarks\n\nThe pivot positions do not depend on the choice of the row echelon form.\nThe number of pivots is the [[rank|0ZC9YTXZRY]] of the matrix.\nIn the system $Ax = b$, the unknowns of the pivot columns are the *basic variables*, and the other unknowns are the *free variables*.\nIn numerical work, partial pivoting chooses as pivot an entry of largest absolute value in its column, to reduce rounding errors.\n\n## Explanations\n\n### L1 human\n\nA pivot is the first nonzero number in a row of a staircase matrix.\nDuring elimination we divide by the pivot and use it to clear the entries below it.\nEach pivot fixes one unknown, and each column without a pivot gives a free unknown.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "defines": "0J0BBCMJ5W",
    "kind": "definition",
    "msc": [
      "15A06"
    ],
    "name": "Pivot",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Pivot_element"
      }
    ],
    "tag": "0BCMGDHZGR",
    "uses": [
      "0F27PBRWD7"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0BCMGDHZGR/",
  "links": [
    {
      "field": "defines",
      "target": "0J0BBCMJ5W"
    },
    {
      "field": "uses",
      "target": "0F27PBRWD7"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0BCMGDHZGR.md",
  "msc": [
    "15A06"
  ],
  "name": "Pivot",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0BC/0BCMGDHZGR.md",
  "status": null,
  "tag": "0BCMGDHZGR",
  "uri": "m2://0BCMGDHZGR",
  "url": "https://math2.tamnd.com/wiki/pivot-definition/",
  "writing": {
    "tag": "0BCMGDHZGR",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}