{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $A$ be an $n \\times n$ complex [[square matrix|0JXZKX526R]].\nThen there are a [[unitary matrix|0AR68JTMDT]] $Q$ and an [[upper triangular matrix|0N4XY7Z3NN]] $T$ such that\n$$A = Q T Q^*,$$\nwhere $Q^*$ is the [[conjugate transpose|0DXKE10PHF]] of $Q$.\nThe diagonal entries of $T$ are the [[eigenvalues|0BQAA2BK3Q]] of $A$, each repeated as often as its algebraic multiplicity.\n\n## Remarks\n\nThe proof is by induction on $n$.\nOver $\\C$, the matrix $A$ has an eigenvector $v$ of norm $1$.\nExtend $v$ to an orthonormal basis; in this basis the first column of the matrix of $A$ is $(\\lambda, 0, \\dots, 0)^T$.\nApply the induction hypothesis to the lower right $(n-1) \\times (n-1)$ block.\n\nThe order of the eigenvalues on the diagonal of $T$ can be chosen freely.\nIf $A$ is a [[normal matrix|0MR0BYF0K2]], then $T$ is normal and triangular, hence diagonal; this gives the [[spectral theorem for normal matrices|04CW1P44VD]].\nThe complex field matters.\nA real matrix with non-real eigenvalues has no factorisation $A = QTQ^T$ with $Q$ real orthogonal and $T$ real upper triangular.\nIndeed, the diagonal of such a $T$ would hold the eigenvalues of $A$.\nThe real Schur form uses an orthogonal $Q$ and a block upper triangular $T$ with diagonal blocks of size $1$ or $2$ instead.\n\n## Explanations\n\n### L1 human\n\nEvery complex square matrix becomes upper triangular after a change to a suitable orthonormal basis.\nThe eigenvalues then sit on the diagonal.\nUnlike diagonalisation, this always works, and the change of basis keeps all lengths and angles.\n\n### L2 human\n\nThe Schur decomposition is the unitary form of [[triangularisation|0363JMQ42Y]].\nIt is stable to compute, and the [[QR algorithm|B6F8YWHYZ5]] for eigenvalues computes it.\nIt gives quick proofs that the trace is the sum and the determinant is the product of the eigenvalues.\nIt is also the standard route to the spectral theorem for normal matrices.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "0AR68JTMDT",
      "0N4XY7Z3NN"
    ],
    "kind": "statement",
    "msc": [
      "15A23",
      "15A21"
    ],
    "name": "Schur decomposition",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q1064218",
      "wikipedia": "Schur decomposition"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Schur_decomposition"
      }
    ],
    "tag": "V1JG4C7R7B",
    "uses": [
      "0JXZKX526R",
      "0AR68JTMDT",
      "0N4XY7Z3NN",
      "0DXKE10PHF",
      "0BQAA2BK3Q"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/V1JG4C7R7B/",
  "links": [
    {
      "field": "about",
      "target": "0AR68JTMDT"
    },
    {
      "field": "about",
      "target": "0N4XY7Z3NN"
    },
    {
      "field": "uses",
      "target": "0JXZKX526R"
    },
    {
      "field": "uses",
      "target": "0AR68JTMDT"
    },
    {
      "field": "uses",
      "target": "0N4XY7Z3NN"
    },
    {
      "field": "uses",
      "target": "0DXKE10PHF"
    },
    {
      "field": "uses",
      "target": "0BQAA2BK3Q"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/V1JG4C7R7B.md",
  "msc": [
    "15A23",
    "15A21"
  ],
  "name": "Schur decomposition",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/V1J/V1JG4C7R7B.md",
  "status": null,
  "tag": "V1JG4C7R7B",
  "uri": "m2://V1JG4C7R7B",
  "url": "https://math2.tamnd.com/wiki/schur-decomposition/",
  "writing": {
    "tag": "V1JG4C7R7B",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}