PlanetMath: orthogonal group

(more info)

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orthogonal group

(Definition)

Let 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 if for all vectors $x,y \in V$ . The subgroup of the general linear group $\operatorname{GL}(V)$ consisting of all linear transformations that preserve is called the orthogonal group with respect to , and denoted $\operatorname{O}(n,Q)$ .

If is also positive definite (i.e., 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 is an orthogonal transformation if and only if $T^{-1} = T^{\operatorname{T}}$ (i.e., the inverse of equals the transpose of ).