{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "RVMW0RP050"
    }
  ],
  "body": "\n## Summary\n\nAn upper Hessenberg matrix is a square matrix whose entries below the first subdiagonal are zero; a lower Hessenberg matrix is its transpose.\nIt is almost [[triangular|GTG8MJGDKC]], with one extra diagonal.\nEvery complex square matrix is unitarily similar to an upper Hessenberg matrix, and this reduction takes finitely many steps.\nFor this reason the [[QR algorithm|B6F8YWHYZ5]] for eigenvalues first brings a matrix to Hessenberg form.\n\n## Explanations\n\n### L1 human\n\nStart from an upper triangular matrix and allow one more diagonal of nonzero entries just below the main diagonal.\nThe result is an upper Hessenberg matrix, such as $\\begin{pmatrix} 1 \u0026 2 \u0026 3 \\\\ 4 \u0026 5 \u0026 6 \\\\ 0 \u0026 7 \u0026 8 \\end{pmatrix}$.\nExact triangular form usually needs the eigenvalues, which are hard to find.\nHessenberg form is the closest one can get with a fixed, finite amount of work, and it makes each later step of eigenvalue algorithms much cheaper.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "15B99",
      "65F15"
    ],
    "name": "Hessenberg matrix",
    "same_as": {
      "wikidata": "Q428813",
      "wikipedia": "Hessenberg matrix"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hessenberg_matrix"
      }
    ],
    "specializes": [
      "0JXZKX526R"
    ],
    "tag": "YCEEWJTX56"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/YCEEWJTX56/",
  "links": [
    {
      "field": "specializes",
      "target": "0JXZKX526R"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/YCEEWJTX56.md",
  "msc": [
    "15B99",
    "65F15"
  ],
  "name": "Hessenberg matrix",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YCE/YCEEWJTX56.md",
  "status": null,
  "tag": "YCEEWJTX56",
  "uri": "m2://YCEEWJTX56",
  "url": "https://math2.tamnd.com/wiki/hessenberg-matrix/",
  "writing": {
    "tag": "YCEEWJTX56",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}