{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $a \u003c b$ and let $K \\in L^2([a, b] \\times [a, b])$ be a [[square-integrable kernel|WWMWX7AD4V]], with [[Hilbert–Schmidt integral operator|CB5G212Q03]] $T_K$ on $L^2[a, b]$ and [[adjoint|6HSJBJ8TTQ]] $T_K^*$.\nThen there are a finite or countable index set $N$ and orthonormal families $(u_n)_{n \\in N}$ and $(v_n)_{n \\in N}$ in $L^2[a, b]$.\nThere are also numbers $s_1 \\ge s_2 \\ge \\cdots \u003e 0$, with $s_n \\to 0$ if $N$ is infinite, such that:\n\n1. $T_K v_n = s_n u_n$ and $T_K^* u_n = s_n v_n$ for every $n \\in N$;\n2. $T_K \\varphi = \\sum_{n \\in N} s_n \\ip{\\varphi}{v_n} u_n$ for every $\\varphi \\in L^2[a, b]$;\n3. $K(x, y) = \\sum_{n \\in N} s_n \\, u_n(x) \\overline{v_n(y)}$, with convergence in $L^2([a, b] \\times [a, b])$;\n4. $\\sum_{n \\in N} s_n^2 = \\int_a^b \\int_a^b \\abs{K(x, y)}^2 \\, dx \\, dy$.\n\nThe numbers $s_n$ are the [[singular values|QSJKPKPM7H]] of $T_K$: their squares are the nonzero eigenvalues of $T_K^* T_K$, counted with multiplicity.\n\n## Remarks\n\nThe operator $T_K^* T_K$ is compact, self-adjoint and has nonnegative eigenvalues; it is the integral operator with the kernel $\\int_a^b \\overline{K(t, x)} K(t, y) \\, dt$.\nThe [[spectral theorem for compact self-adjoint operators|A1G25R99PY]] gives orthonormal eigenfunctions $v_n$ with eigenvalues $s_n^2 \u003e 0$.\nThen $u_n = T_K v_n / s_n$ is an orthonormal family, since $\\ip{T_K v_n}{T_K v_m} = \\ip{T_K^* T_K v_n}{v_m} = s_n^2 \\delta_{nm}$.\nParts 3 and 4 follow as for the bilinear series of a symmetric kernel, using the orthonormal family $u_n(x) \\overline{v_m(y)}$.\n\nExample: the Volterra kernel on $[0, 1]$, equal to $1$ for $y \\le x$ and $0$ otherwise, has no eigenvalues, but it has singular values $s_n = \\frac{1}{(n - \\frac12)\\pi}$, with $v_n(y) = \\sqrt{2} \\cos\\left((n - \\tfrac12)\\pi y\\right)$ and $u_n(x) = \\sqrt{2} \\sin\\left((n - \\tfrac12)\\pi x\\right)$.\nPart 4 gives $\\sum_n \\frac{1}{(n - \\frac12)^2 \\pi^2} = \\frac12$, the area of the triangle where the kernel is $1$.\n\nFor a Hermitian kernel, the $s_n$ are the absolute values of the nonzero eigenvalues, and one may take $u_n = \\pm v_n$.\n\n## Explanations\n\n### L1 human\n\nEvery square-integrable kernel, symmetric or not, can be written as a sum of simple products $u_n(x) \\overline{v_n(y)}$ with decreasing positive weights.\nThe two families of functions are each orthonormal, but they are different families.\nThis is the integral version of the [[singular value decomposition|000NPTMX6J]] of a matrix.\n\n### L2 human\n\nThe decomposition is the right tool for non-symmetric kernels, where eigenfunctions may fail to exist or to be orthogonal.\nKeeping the first $m$ terms gives the best approximation of $K$ in the $L^2$ norm by a degenerate kernel of rank at most $m$, with error $\\left(\\sum_{n \u003e m} s_n^2\\right)^{1/2}$.\nFor a Fredholm equation of the first kind $T_K \\varphi = f$, it gives the formal solution $\\varphi = \\sum_n s_n^{-1} \\ip{f}{u_n} v_n$, and the growth of $1/s_n$ shows why such equations are ill-posed.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "WWMWX7AD4V",
      "QSJKPKPM7H"
    ],
    "kind": "statement",
    "msc": [
      "45C05",
      "47B10",
      "15A18"
    ],
    "name": "Schmidt's singular value decomposition of a kernel",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Singular_value_decomposition"
      }
    ],
    "tag": "W4E25FVXV1",
    "uses": [
      "WWMWX7AD4V",
      "CB5G212Q03",
      "6HSJBJ8TTQ",
      "QSJKPKPM7H",
      "A1G25R99PY",
      "000NPTMX6J"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/W4E25FVXV1/",
  "links": [
    {
      "field": "about",
      "target": "WWMWX7AD4V"
    },
    {
      "field": "about",
      "target": "QSJKPKPM7H"
    },
    {
      "field": "uses",
      "target": "WWMWX7AD4V"
    },
    {
      "field": "uses",
      "target": "CB5G212Q03"
    },
    {
      "field": "uses",
      "target": "6HSJBJ8TTQ"
    },
    {
      "field": "uses",
      "target": "QSJKPKPM7H"
    },
    {
      "field": "uses",
      "target": "A1G25R99PY"
    },
    {
      "field": "uses",
      "target": "000NPTMX6J"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/W4E25FVXV1.md",
  "msc": [
    "45C05",
    "47B10",
    "15A18"
  ],
  "name": "Schmidt's singular value decomposition of a kernel",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/W4E/W4E25FVXV1.md",
  "status": null,
  "tag": "W4E25FVXV1",
  "uri": "m2://W4E25FVXV1",
  "url": "https://tamnd.github.io/math2/wiki/schmidts-singular-value-decomposition-of-a-kernel/",
  "writing": {
    "tag": "W4E25FVXV1",
    "kind": "statement",
    "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
      }
    ]
  }
}