{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0B9QNFGTHM"
    },
    {
      "field": "uses",
      "target": "0TKYAEJ8N7"
    },
    {
      "field": "uses",
      "target": "0VMWGYY35Q"
    }
  ],
  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $Q$ be a real $n \\times n$ matrix.\nThe matrix $Q$ is *orthogonal* when\n$$Q^T Q = I_n,$$\nwhere $Q^T$ is the [[transpose|09DXWYRD43]] and $I_n$ is the [[identity matrix|06DW9BAWB1]].\n\n## Remarks\n\nEquivalently, $Q$ is invertible with $Q^{-1} = Q^T$, and then also $Q Q^T = I_n$.\nEquivalently, the columns of $Q$ form an orthonormal basis of $\\R^n$; the same holds for the rows.\nThe map $x \\mapsto Qx$ keeps the dot product, so it is a linear isometry of $\\R^n$.\nThe determinant of an orthogonal matrix is $1$ or $-1$.\n\n## Explanations\n\n### L1 human\n\nOrthogonal matrices describe the rigid motions of space that fix the origin: rotations and reflections.\nThey keep all lengths and angles.\nTo invert an orthogonal matrix, we only need to transpose it.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0W02QESPJQ",
    "kind": "definition",
    "msc": [
      "15B10"
    ],
    "name": "Orthogonal matrix",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Orthogonal_matrix"
      }
    ],
    "tag": "0QRBVFJ50X",
    "uses": [
      "0W809K6AW0",
      "06DW9BAWB1"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0QRBVFJ50X/",
  "links": [
    {
      "field": "defines",
      "target": "0W02QESPJQ"
    },
    {
      "field": "uses",
      "target": "0W809K6AW0"
    },
    {
      "field": "uses",
      "target": "06DW9BAWB1"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0QRBVFJ50X.md",
  "msc": [
    "15B10"
  ],
  "name": "Orthogonal matrix",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0QR/0QRBVFJ50X.md",
  "status": null,
  "tag": "0QRBVFJ50X",
  "uri": "m2://0QRBVFJ50X",
  "url": "https://math2.tamnd.com/wiki/orthogonal-matrix-definition/",
  "writing": {
    "tag": "0QRBVFJ50X",
    "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
      }
    ]
  }
}