{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $a \u003c b$.\nA *symmetric kernel* on $[a, b]$ is a real-valued [[kernel|GSMBXT7FPK]] $K \\colon [a, b] \\times [a, b] \\to \\R$ such that\n$$K(x, s) = K(s, x) \\quad \\text{for all } x, s \\in [a, b].$$\n\n## Remarks\n\nExample: $K(x, s) = \\min(x, s)$ on $[0, 1]$.\nThe check: the minimum of two numbers does not depend on their order.\n\nNon-example: $K(x, s) = x - s$.\nHere $K(1, 0) = 1$ and $K(0, 1) = -1$.\n\nA symmetric kernel is a Hermitian kernel that takes real values.\nFor square-integrable kernels the condition is often required only for almost all $(x, s)$.\nSome authors call a complex kernel with $K(x, s) = K(s, x)$ symmetric as well; that condition is different from the Hermitian one.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "ADTYF8FG6C",
    "kind": "definition",
    "msc": [
      "45C05",
      "45P05"
    ],
    "name": "Symmetric kernel",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hilbert%E2%80%93Schmidt_integral_operator"
      }
    ],
    "tag": "XY7H05ANPD",
    "uses": [
      "GSMBXT7FPK"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/XY7H05ANPD/",
  "links": [
    {
      "field": "defines",
      "target": "ADTYF8FG6C"
    },
    {
      "field": "uses",
      "target": "GSMBXT7FPK"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/XY7H05ANPD.md",
  "msc": [
    "45C05",
    "45P05"
  ],
  "name": "Symmetric kernel",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/XY7/XY7H05ANPD.md",
  "status": null,
  "tag": "XY7H05ANPD",
  "uri": "m2://XY7H05ANPD",
  "url": "https://tamnd.github.io/math2/wiki/symmetric-kernel-definition/",
  "writing": {
    "tag": "XY7H05ANPD",
    "kind": "definition",
    "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
      }
    ]
  }
}