# 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.