{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 1$.\nA function $f \\colon \\R^n \\to \\C$ is *positive-definite* if for every $N \\ge 1$, all points $x_1, \\dots, x_N \\in \\R^n$ and all $c_1, \\dots, c_N \\in \\C$,\n$$\\sum_{i=1}^{N} \\sum_{j=1}^{N} c_i \\, \\overline{c_j} \\, f(x_i - x_j) \\ge 0.$$\n\n## Remarks\n\nEquivalently, each matrix $\\big( f(x_i - x_j) \\big)_{i,j=1}^{N}$ is [[Hermitian|0EKZCX4K3P]] and [[positive semidefinite|0880F9EQCP]].\nTaking $N = 1$ and $N = 2$ gives $f(0) \\ge 0$, $f(-x) = \\overline{f(x)}$ and $\\abs{f(x)} \\le f(0)$.\nThis is the case $G = \\R^n$ of a [[positive-definite function on a group|3PRZ02APFA]], except that continuity is not part of the definition here.\nSome authors call these functions positive semidefinite and reserve \"positive-definite\" for strict inequality when the points are distinct and $c \\ne 0$.\n\nExample: $f(x) = \\cos x$ on $\\R$.\nSince $\\cos(x_i - x_j) = \\frac12 \\big( e^{ix_i} e^{-ix_j} + e^{-ix_i} e^{ix_j} \\big)$,\n$$\\sum_{i,j} c_i \\overline{c_j} \\cos(x_i - x_j) = \\frac12 \\Big| \\sum_i c_i e^{ix_i} \\Big|^2 + \\frac12 \\Big| \\sum_i c_i e^{-ix_i} \\Big|^2 \\ge 0.$$\n\nNon-example: $f = 1_{[-1,1]}$ on $\\R$.\nFor the points $0, 1, 2$ the matrix $\\big(f(x_i - x_j)\\big)$ is\n$$\\begin{pmatrix} 1 \u0026 1 \u0026 0 \\\\ 1 \u0026 1 \u0026 1 \\\\ 0 \u0026 1 \u0026 1 \\end{pmatrix},$$\nwhose determinant is $-1$.\nA positive semidefinite matrix has nonnegative determinant, so $f$ is not positive-definite.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "F73VWGF637",
    "kind": "definition",
    "msc": [
      "42A82",
      "43A35"
    ],
    "name": "Positive-definite function",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Positive-definite_function"
      }
    ],
    "tag": "VPMHPRRKZH",
    "uses": [
      "0880F9EQCP",
      "0EKZCX4K3P"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/VPMHPRRKZH/",
  "links": [
    {
      "field": "defines",
      "target": "F73VWGF637"
    },
    {
      "field": "uses",
      "target": "0880F9EQCP"
    },
    {
      "field": "uses",
      "target": "0EKZCX4K3P"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/VPMHPRRKZH.md",
  "msc": [
    "42A82",
    "43A35"
  ],
  "name": "Positive-definite function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VPM/VPMHPRRKZH.md",
  "status": null,
  "tag": "VPMHPRRKZH",
  "uri": "m2://VPMHPRRKZH",
  "url": "https://tamnd.github.io/math2/wiki/positive-definite-function-definition/",
  "writing": {
    "tag": "VPMHPRRKZH",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}