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Orthogonal group

An orthogonal group is any member of a sequence $(O_N)_{N\in \N}$ of classical matrix groups.

Definition

For every $N\in \N$ the orthogonal group for dimension $N$ is the subgroup of the general linear group $\mathrm{GL}(N,\C)$ given by all orthogonal $N\times N$ -matrices with complex entries, i.e., the set

$O_N\colon\hspace{-0.6em}=\{ u\in \C^{N\times N}\,\vert\, u=\overline{u},\, uu^t=u^tu=I\},$

where, if $u=(u_{i,j})_{i,j=1}^N$ , then $\overline u=(u^\ast_{i,j})_{i,j=1}^N$ is the complex conjugate of $u$ and $u^t=(u_{j,i})_{i,j=1}^N$ the transpose and where $I_N$ is the identity $N\!\times \!N$ -matrix.