{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]].\nLet $n \\ge 1$, let $z_1, \\ldots, z_n \\in \\mathbb{D}$ be distinct, and let $w_1, \\ldots, w_n \\in \\C$.\nThe *Pick matrix* of the data $(z_i, w_i)$ is the $n \\times n$ [[matrix|0N1MY1T0W9]] $P$ with entries\n$$P_{ij} = \\frac{1 - w_i \\overline{w_j}}{1 - z_i \\overline{z_j}}, \\qquad 1 \\le i, j \\le n.$$\n\n## Remarks\n\nThe denominators are not $0$, because $\\abs{z_i \\overline{z_j}} \u003c 1$.\nOne has $\\overline{P_{ji}} = P_{ij}$, so $P$ is a [[Hermitian matrix|0EKZCX4K3P]].\nBy [[Nevanlinna–Pick interpolation|4VMSSYB5DN]], with all $w_i \\in \\mathbb{D}$, there is a holomorphic $f \\colon \\mathbb{D} \\to \\mathbb{D}$ with $f(z_i) = w_i$ if and only if $P$ is [[positive semidefinite|0880F9EQCP]].\n\nExample: $n = 2$, $z_1 = 0$, $w_1 = 0$, $z_2 = a$, $w_2 = b$ with $a \\ne 0$.\nThen $P = \\begin{pmatrix} 1 \u0026 1 \\\\ 1 \u0026 \\frac{1 - \\abs{b}^2}{1 - \\abs{a}^2} \\end{pmatrix}$, and $\\det P = \\frac{\\abs{a}^2 - \\abs{b}^2}{1 - \\abs{a}^2}$.\nSo $P$ is positive semidefinite exactly when $\\abs{b} \\le \\abs{a}$, which matches the [[Schwarz lemma|0MB48XDBY3]].\n\nNon-example: the data must have $z_i \\in \\mathbb{D}$.\nFor $z_1 = 1$ on the unit circle the entry $P_{11}$ would have denominator $1 - \\abs{z_1}^2 = 0$, so no Pick matrix is defined.\n\nThe entries are the values $(1 - w_i \\overline{w_j}) k(z_i, z_j)$, where $k(z, w) = \\frac{1}{1 - z \\bar w}$ is the reproducing kernel of the [[Hardy space|PNNNFJVF6W]] $H^2$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "TSPTN464CM",
    "kind": "definition",
    "msc": [
      "30E05",
      "47A57"
    ],
    "name": "Pick matrix",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Nevanlinna–Pick_interpolation"
      }
    ],
    "tag": "73X2JGSE0W",
    "uses": [
      "8G5ABXRK85",
      "0EKZCX4K3P",
      "0N1MY1T0W9"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/73X2JGSE0W/",
  "links": [
    {
      "field": "defines",
      "target": "TSPTN464CM"
    },
    {
      "field": "uses",
      "target": "8G5ABXRK85"
    },
    {
      "field": "uses",
      "target": "0EKZCX4K3P"
    },
    {
      "field": "uses",
      "target": "0N1MY1T0W9"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/73X2JGSE0W.md",
  "msc": [
    "30E05",
    "47A57"
  ],
  "name": "Pick matrix",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/73X/73X2JGSE0W.md",
  "status": null,
  "tag": "73X2JGSE0W",
  "uri": "m2://73X2JGSE0W",
  "url": "https://math2.tamnd.com/wiki/pick-matrix-definition/",
  "writing": {
    "tag": "73X2JGSE0W",
    "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
      }
    ]
  }
}