{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "4W0EG94RZ5"
    },
    {
      "field": "uses",
      "target": "4W0EG94RZ5"
    },
    {
      "field": "uses",
      "target": "9NZ9CXC8FY"
    },
    {
      "field": "defines",
      "target": "WKM1B317YB"
    }
  ],
  "body": "\n## Summary\n\nThe Fredholm minors of a continuous kernel $K$ are entire functions of the parameter $\\lambda$.\nThey are built like the [[Fredholm determinant|PPX0AK81QV]], from determinants of values of $K$, but with some rows and columns held at fixed points.\nThey are the integral analogues of the minors and cofactors of the matrix $I - \\lambda A$.\nThe first minor divided by the Fredholm determinant is the [[resolvent kernel|ARTJV1XCRX]].\nSo the minors give the explicit solution of the [[Fredholm equation of the second kind|DEYYNVCV9J]].\nAt a zero of the determinant, the first minor that is not identically zero yields the solutions of the homogeneous equation.\n\n## Explanations\n\n### L1 human\n\nFor a square matrix, Cramer's rule writes the inverse as cofactors divided by the determinant.\nFredholm minors are the cofactors of an integral operator.\nThe first minor $D(x, y; \\lambda)$ depends on two points $x$ and $y$, and $D(x, y; \\lambda) / D(\\lambda)$ is the kernel of the inverse operation.\nFor $K(x, y) = xy$ on $[0, 1]$ one finds $D(x, y; \\lambda) = xy$, so the resolvent kernel is $xy / (1 - \\lambda/3)$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "45B05"
    ],
    "name": "Fredholm minors",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fredholm_theory"
      }
    ],
    "tag": "HP5VJ9EBST"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/HP5VJ9EBST/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/HP5VJ9EBST.md",
  "msc": [
    "45B05"
  ],
  "name": "Fredholm minors",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HP5/HP5VJ9EBST.md",
  "status": null,
  "tag": "HP5VJ9EBST",
  "uri": "m2://HP5VJ9EBST",
  "url": "https://tamnd.github.io/math2/wiki/fredholm-minors/",
  "writing": {
    "tag": "HP5VJ9EBST",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}