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 $T$ ).

单词	OrthogonalGroup
释义	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 $T$ ).
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