Differences

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--- modified_bistochastic_group [2020/02/14 07:11]
amang [Basic properties]
+++ modified_bistochastic_group [2021/11/23 11:56] (current)
@@ Line 1: / Line 1: @@
 ====== Modified bistochastic group ======
-A **modified bistochastic group** is any member of a certain sequence $(B_N')_{N\in \N}$ of [[classical matrix groups]].
+A **modified bistochastic group** is any member of a certain sequence $(B_N')_{N\in \N}$ of [[classical orthogonal matrix groups]].
 ===== Definition =====
@@ Line 14: / Line 14: @@
 For every $N\in \N$ the **modified bistochastic group** for dimension $N$ is the subgroup of the [[wp>general linear group]] $\mathrm{GL}(N,\C)$ given by all bistochastic [[orthogonal group|orthogonal]] $N\times N$-matrices multiplied by a factor of $1$ or $-1$, i.e., the set
 $$B_N'\colon\hspace{-0.6em}=\{ u\in \C^{N\times N}\,\vert\, u=\overline{u},\, uu^t=u^tu=I,\,\exists r\in\{-1,1\}: \forall_{i,j=1}^N: {\textstyle\sum_{\ell=1}^N} u_{i,\ell}={\textstyle\sum_{k=1}^N} u_{k,j}=r\},$$
-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.