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.

Also see

 * Orthogonal Group is Group
 * Orthogonal Group is Subgroup of General Linear Group