Let $Q$ be a non-degenerate symmetric bilinear form over the real vector space $\mathbb{R}^n$ A linear transformation $T\colon V \to V$ is said to preserve $Q$ if $Q(Tx,Ty) = Q(x,y)$ for all vectors $x,y \in V$ The subgroup of the general linear group $\operatorname{GL}(V)$ consisting of all linear transformations that preserve $Q$ is called the orthogonal group with respect to $Q$ and denoted $\operatorname{O}(n,Q)$

If $Q$ is also positive definite (i.e., $Q$ is an inner product), then $\operatorname{O}(n,Q)$ is equivalent to the group of invertible linear transformations that preserve the standard inner product on $\mathbb{R}^n$ and in this case the group $\operatorname{O}(n,Q)$ is usually denoted $\operatorname{O}(n)$

Elements of $\operatorname{O}(n)$ are called orthogonal transformations. One can show that a linear transformation $T$ is an orthogonal transformation if and only if $T^{-1} = T^{\operatorname{T}}$ (i.e., the inverse of $T$ equals the transpose of $T$ .