Definition:Orthogonal Group

Definition
Let $k$ be a field.

The ($n$th) orthogonal group (on $k$), denoted $\operatorname O \left({n, k}\right)$, is the following subset of the general linear group $\operatorname{GL} \left({n, k}\right)$:


 * $\operatorname O \left({n, k}\right) := \left\{ {M \in \operatorname{GL} \left({n, k}\right): M^\intercal = M^{-1} }\right\}$

where $M^\intercal$ denotes the transpose of $M$.

Further, $\operatorname O \left({n, k}\right)$ is considered to be endowed with conventional matrix multiplication.

That is, the ($n$th) orthogonal group (on $k$) is the set of all orthogonal order-$n$ square matrices over $k$ under (conventional) matrix multiplication.

Orthogonal Group of Bilinear Form
Let $V$ be a vector space over a field $\mathbb K$.

Let $B : V\times V\to \mathbb K$ be a nondegenerate bilinear form.

Its orthogonal group $O(B)$ is the group of invertible linear transformations $g\in\operatorname{GL}(V)$ such that:
 * $\forall v,w\in V : B(gv, gw) = B(v,w) $

Orthogonal Group of Inner Product Space
Let $V$ be an inner product space.

Its orthogonal group $O(V)$ is the group of invertible linear transformations $g\in\operatorname{GL}(V)$ such that:
 * $\forall v,w\in V : \langle gv, gw\rangle = \langle v,w\rangle $

That is, it is the orthogonal group of its inner product.

Also see

 * Definition:Special Orthogonal Group
 * Definition:Unitary Group
 * Orthogonal Group is Group
 * Orthogonal Group is Subgroup of General Linear Group