{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0880F9EQCP"
    },
    {
      "field": "uses",
      "target": "0DGZBQWZWN"
    },
    {
      "field": "uses",
      "target": "0K4WCNTMHG"
    },
    {
      "field": "uses",
      "target": "0QQCG3JWJ4"
    }
  ],
  "body": "\n## Definition\n\nLet $A = (a_{ij})$ be an $n \\times n$ complex [[square matrix|0JXZKX526R]].\nThe matrix $A$ is *Hermitian* when $A^* = A$, where $A^*$ is the [[conjugate transpose|0DXKE10PHF]] of $A$.\nEquivalently, $a_{ij} = \\overline{a_{ji}}$ for all $i$ and $j$.\n\n## Remarks\n\nThe diagonal entries of a Hermitian matrix are real.\nA real matrix is Hermitian if and only if it is [[symmetric|0VR671PHE5]].\nThe eigenvalues of a Hermitian matrix are real, and $\\C^n$ has an orthonormal basis of eigenvectors, by the [[spectral theorem for Hermitian matrices|0DGZBQWZWN]].\nThe matrix $\\begin{pmatrix} 2 \u0026 1 - i \\\\ 1 + i \u0026 3 \\end{pmatrix}$ is Hermitian.\n\n## Explanations\n\n### L1 human\n\nA Hermitian matrix is the complex version of a symmetric matrix.\nThe entry across the diagonal is the complex conjugate of the entry here.\nIn quantum mechanics, observable quantities are Hermitian matrices or operators.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0P17NGQQMQ",
    "kind": "definition",
    "msc": [
      "15B57"
    ],
    "name": "Hermitian matrix",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Hermitian_matrix"
      }
    ],
    "tag": "0EKZCX4K3P",
    "uses": [
      "0EX045E2BP",
      "0CP7YY6NAG"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0EKZCX4K3P/",
  "links": [
    {
      "field": "defines",
      "target": "0P17NGQQMQ"
    },
    {
      "field": "uses",
      "target": "0EX045E2BP"
    },
    {
      "field": "uses",
      "target": "0CP7YY6NAG"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0EKZCX4K3P.md",
  "msc": [
    "15B57"
  ],
  "name": "Hermitian matrix",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0EK/0EKZCX4K3P.md",
  "status": null,
  "tag": "0EKZCX4K3P",
  "uri": "m2://0EKZCX4K3P",
  "url": "https://math2.tamnd.com/wiki/hermitian-matrix-definition/",
  "writing": {
    "tag": "0EKZCX4K3P",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}