{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] and let $A = (a_{ij})$ be an $n \\times n$ [[square matrix|0JXZKX526R]] over $R$.\nThe matrix $A$ is *upper Hessenberg* when $a_{ij} = 0$ for all $i \u003e j + 1$.\nIt is *lower Hessenberg* when $a_{ij} = 0$ for all $j \u003e i + 1$.\n\n## Remarks\n\nExample: $\\begin{pmatrix} 1 \u0026 2 \u0026 3 \\\\ 4 \u0026 5 \u0026 6 \\\\ 0 \u0026 7 \u0026 8 \\end{pmatrix}$ is upper Hessenberg, because its only entry with $i \u003e j + 1$ is $a_{31} = 0$.\nNon-example: $\\begin{pmatrix} 1 \u0026 2 \u0026 3 \\\\ 4 \u0026 5 \u0026 6 \\\\ 7 \u0026 8 \u0026 9 \\end{pmatrix}$ is not upper Hessenberg, because $a_{31} = 7 \\ne 0$.\nEvery [[upper triangular matrix|0N4XY7Z3NN]] is upper Hessenberg.\nA matrix that is both upper and lower Hessenberg is a [[tridiagonal matrix|DARX1KZRHZ]].\nAn upper Hessenberg matrix is *unreduced* when all its subdiagonal entries $a_{i+1, i}$ are nonzero.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "YCEEWJTX56",
    "kind": "definition",
    "msc": [
      "15B99",
      "65F15"
    ],
    "name": "Hessenberg matrix",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hessenberg_matrix"
      }
    ],
    "tag": "RVMW0RP050",
    "uses": [
      "0B1261B5ED",
      "0JXZKX526R"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/RVMW0RP050/",
  "links": [
    {
      "field": "defines",
      "target": "YCEEWJTX56"
    },
    {
      "field": "uses",
      "target": "0B1261B5ED"
    },
    {
      "field": "uses",
      "target": "0JXZKX526R"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/RVMW0RP050.md",
  "msc": [
    "15B99",
    "65F15"
  ],
  "name": "Hessenberg matrix",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RVM/RVMW0RP050.md",
  "status": null,
  "tag": "RVMW0RP050",
  "uri": "m2://RVMW0RP050",
  "url": "https://math2.tamnd.com/wiki/hessenberg-matrix-definition/",
  "writing": {
    "tag": "RVMW0RP050",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}