orthogonal group

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 $T$ ).

Title	orthogonal group
Canonical name	OrthogonalGroup
Date of creation	2013-03-22 12:25:54
Last modified on	2013-03-22 12:25:54
Owner	djao (24)
Last modified by	djao (24)
Numerical id	6
Author	djao (24)
Entry type	Definition
Classification	msc 20G20
Defines	orthogonal transformation