---
tag: M2900DMEW6
kind: example
name: Indefinite orthogonal group
msc: ["22E15", "20G20", "15A63"]
instance_of: [7JBRGCVJRB]
uses: [0TKYAEJ8N7, 0RBQS1JJCR, 07EZA80NEH]
sources:
  - url: https://en.wikipedia.org/wiki/Indefinite_orthogonal_group
same_as:
  wikidata: Q11061828
  wikipedia: Indefinite orthogonal group
---

## Example

Let $p, q \ge 0$ be integers with $n = p + q \ge 1$, and let
$$\eta = \begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix}.$$
The 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]] $(p, q)$.
The *indefinite orthogonal group* is
$$O(p, q) = \set{A \in GL_n(\R) : A^T \eta A = \eta},$$
the group of linear maps of $\R^n$ that preserve $Q$.
It is a closed subgroup of $GL_n(\R)$, so it is a [[Lie group|7JBRGCVJRB]].

Check with $p = q = 1$: for every $t \in \R$, the matrix
$$A_t = \begin{pmatrix} \cosh t & \sinh t \\ \sinh t & \cosh t \end{pmatrix}$$
satisfies $A_t^T \eta A_t = \begin{pmatrix} \cosh^2 t - \sinh^2 t & 0 \\ 0 & \sinh^2 t - \cosh^2 t \end{pmatrix} = \eta$, so $A_t \in O(1, 1)$.
Its entries are unbounded as $t \to \infty$, so $O(1, 1)$ is not compact.

## Remarks

For $q = 0$ the group $O(n, 0)$ is the compact [[orthogonal group|0TKYAEJ8N7]] $O(n)$.
The dimension of $O(p, q)$ is $n(n - 1)/2$, as for $O(n)$.
$O(p, q)$ and $O(q, p)$ are isomorphic, since replacing $\eta$ by $-\eta$ does not change the group.
When $p$ and $q$ are both positive, $O(p, q)$ is not compact and has four connected components.
Its subgroup $SO(p, q)$ of elements with determinant $1$ has two components, and the identity component is written $SO^+(p, q)$.
A maximal compact subgroup of $O(p, q)$ is $O(p) \times O(q)$.
The Lorentz group of special relativity is $O(1, 3)$.

## Explanations

### L1 human

The ordinary orthogonal group keeps the usual squared length $x_1^2 + \cdots + x_n^2$.
The indefinite orthogonal group keeps a "length" in which some squares count with a minus sign.
Then the circles of constant length become hyperbolas, and the rotations become hyperbolic rotations built from $\cosh$ and $\sinh$.
These maps can stretch vectors arbitrarily far along the hyperbolas, so the group is no longer compact.