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  "body": "\n## Example\n\nLet $p, q \\ge 0$ be integers with $n = p + q \\ge 1$, and let\n$$\\eta = \\begin{pmatrix} I_p \u0026 0 \\\\ 0 \u0026 -I_q \\end{pmatrix}.$$\nThe matrix $\\eta$ defines the [[quadratic form|0RBQS1JJCR]] $Q(x) = x^T \\eta x = x_1^2 + \\cdots + x_p^2 - x_{p+1}^2 - \\cdots - x_n^2$ on $\\R^n$, of [[signature|07EZA80NEH]] $(p, q)$.\nThe *indefinite orthogonal group* is\n$$O(p, q) = \\set{A \\in GL_n(\\R) : A^T \\eta A = \\eta},$$\nthe group of linear maps of $\\R^n$ that preserve $Q$.\nIt is a closed subgroup of $GL_n(\\R)$, so it is a [[Lie group|7JBRGCVJRB]].\n\nCheck with $p = q = 1$: for every $t \\in \\R$, the matrix\n$$A_t = \\begin{pmatrix} \\cosh t \u0026 \\sinh t \\\\ \\sinh t \u0026 \\cosh t \\end{pmatrix}$$\nsatisfies $A_t^T \\eta A_t = \\begin{pmatrix} \\cosh^2 t - \\sinh^2 t \u0026 0 \\\\ 0 \u0026 \\sinh^2 t - \\cosh^2 t \\end{pmatrix} = \\eta$, so $A_t \\in O(1, 1)$.\nIts entries are unbounded as $t \\to \\infty$, so $O(1, 1)$ is not compact.\n\n## Remarks\n\nFor $q = 0$ the group $O(n, 0)$ is the compact [[orthogonal group|0TKYAEJ8N7]] $O(n)$.\nThe dimension of $O(p, q)$ is $n(n - 1)/2$, as for $O(n)$.\n$O(p, q)$ and $O(q, p)$ are isomorphic, since replacing $\\eta$ by $-\\eta$ does not change the group.\nWhen $p$ and $q$ are both positive, $O(p, q)$ is not compact and has four connected components.\nIts subgroup $SO(p, q)$ of elements with determinant $1$ has two components, and the identity component is written $SO^+(p, q)$.\nA maximal compact subgroup of $O(p, q)$ is $O(p) \\times O(q)$.\nThe Lorentz group of special relativity is $O(1, 3)$.\n\n## Explanations\n\n### L1 human\n\nThe ordinary orthogonal group keeps the usual squared length $x_1^2 + \\cdots + x_n^2$.\nThe indefinite orthogonal group keeps a \"length\" in which some squares count with a minus sign.\nThen the circles of constant length become hyperbolas, and the rotations become hyperbolic rotations built from $\\cosh$ and $\\sinh$.\nThese maps can stretch vectors arbitrarily far along the hyperbolas, so the group is no longer compact.\n",
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