Definition:Orthogonal Group

Definition

Let $k$ be a field.

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

$\map {\mathrm O} {n, k} := \set {M \in \GL {n, k}: M^\intercal = M^{-1} }$

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

Further, $\map {\mathrm O} {n, k}$ 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 $\map {\mathrm O} B$ is the group of invertible linear transformations $g \in \GL V$ such that:

$\forall v, w \in V : \map B {g v, g w} = \map B {v, w}$

Orthogonal Group of Inner Product Space

Let $V$ be an inner product space.

Its orthogonal group $\map {\mathrm O} V$ is the group of invertible linear transformations $g \in \GL V$ such that:

$\forall v, w \in V: \innerprod {g v} {g w} = \innerprod v w$

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