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--- orthogonal_group [2020/02/14 07:08]
amang [Basic properties]
+++ orthogonal_group [2021/11/23 11:56] (current)
@@ Line 1: / Line 1: @@
 ====== Orthogonal group ======
-An **orthogonal group** is any member of a sequence $(O_N)_{N\in \N}$ of [[classical matrix groups]].
+An **orthogonal group** is any member of a sequence $(O_N)_{N\in \N}$ of [[classical orthogonal matrix groups]].
 ===== Definition =====
@@ Line 7: / Line 7: @@
 For every $N\in \N$ the **orthogonal group** for dimension $N$ is the subgroup of the [[wp>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.
+where, if $u=(u_{i,j})_{i,j=1}^N$, then $\overline u=(\overline{u}_{i,j})_{i,j=1}^N$ 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.