{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] and let $A = (a_{ij})$ be an $m \\times n$ [[matrix|0VD9ZSWWJG]] over $R$.\nThe matrix $A$ is a *Hankel matrix* when $a_{i, j+1} = a_{i+1, j}$ for all $1 \\le i \u003c m$ and $1 \\le j \u003c n$.\nEquivalently, there are elements $h_k \\in R$, for $2 \\le k \\le m + n$, with $a_{ij} = h_{i+j}$ for all $i, j$.\n\n## Remarks\n\nExample: $\\begin{pmatrix} 1 \u0026 1 \u0026 2 \\\\ 1 \u0026 2 \u0026 3 \\\\ 2 \u0026 3 \u0026 5 \\end{pmatrix}$ is Hankel, with $h_2, \\dots, h_6 = 1, 1, 2, 3, 5$; each anti-diagonal is constant.\nNon-example: $\\begin{pmatrix} 1 \u0026 2 \\\\ 3 \u0026 4 \\end{pmatrix}$ is not Hankel, because $a_{12} = 2 \\ne 3 = a_{21}$.\nA square Hankel matrix is symmetric, since $a_{ij} = h_{i+j} = a_{ji}$.\nThe [[Hilbert matrix|2DTWB6F5TA]], with $h_k = 1/(k-1)$, is a Hankel matrix.\nIf $J$ is the $m \\times m$ matrix with ones on the anti-diagonal and zeros elsewhere, then $A$ is Hankel exactly when $JA$ is a [[Toeplitz matrix|GKY2WSHE26]].\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "PQ1D8FQB3E",
    "kind": "definition",
    "msc": [
      "15B05"
    ],
    "name": "Hankel matrix",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hankel_matrix"
      }
    ],
    "tag": "TRRREQX56B",
    "uses": [
      "0B1261B5ED",
      "0VD9ZSWWJG"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/TRRREQX56B/",
  "links": [
    {
      "field": "defines",
      "target": "PQ1D8FQB3E"
    },
    {
      "field": "uses",
      "target": "0B1261B5ED"
    },
    {
      "field": "uses",
      "target": "0VD9ZSWWJG"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/TRRREQX56B.md",
  "msc": [
    "15B05"
  ],
  "name": "Hankel matrix",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TRR/TRRREQX56B.md",
  "status": null,
  "tag": "TRRREQX56B",
  "uri": "m2://TRRREQX56B",
  "url": "https://math2.tamnd.com/wiki/hankel-matrix-definition/",
  "writing": {
    "tag": "TRRREQX56B",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}