bistochastic_group
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bistochastic_group [2020/02/14 07:10] amang [Basic properties] |
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| ====== Bistochastic group ====== | ====== Bistochastic group ====== | ||
| - | A **bistochastic group** is any member of a sequence $(B_N)_{N\in \N}$ of [[classical matrix groups]]. | + | A **bistochastic group** is any member of a sequence $(B_N)_{N\in \N}$ of [[classical orthogonal matrix groups]]. |
| ===== Definition ===== | ===== Definition ===== | ||
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| For every $N\in \N$ the **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, i.e., the set | For every $N\in \N$ the **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, i.e., the set | ||
| $$B_N\colon\hspace{-0.6em}=\{ u\in \C^{N\times N}\,\vert\, u=\overline{u},\, uu^t=u^tu=I, \, \forall_{i,j=1}^N: {\textstyle\sum_{\ell=1}^N} u_{i,\ell}={\textstyle\sum_{k=1}^N} u_{k,j}=1\},$$ | $$B_N\colon\hspace{-0.6em}=\{ u\in \C^{N\times N}\,\vert\, u=\overline{u},\, uu^t=u^tu=I, \, \forall_{i,j=1}^N: {\textstyle\sum_{\ell=1}^N} u_{i,\ell}={\textstyle\sum_{k=1}^N} u_{k,j}=1\},$$ | ||
| - | 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. |
| Note that the elements of $B_N$ are __not__ required to have non-negative entries. Stochastic matrices with non-negative entries are known as **probability matrices**, **transition matrices**, **substitution matrices** or **Markov matrices**. The set of such $N\times N$-matrices forms a compact Hausdorff semigroup with respect to the topology inherited from $\C^{N\times N}$ -- but not a group. In fact, a bistochastic matrix with non-negative entries has a bistochastic inverse with non-negative entries if and only if it is a permutation matrix [(:ref:MonPle73)]. | Note that the elements of $B_N$ are __not__ required to have non-negative entries. Stochastic matrices with non-negative entries are known as **probability matrices**, **transition matrices**, **substitution matrices** or **Markov matrices**. The set of such $N\times N$-matrices forms a compact Hausdorff semigroup with respect to the topology inherited from $\C^{N\times N}$ -- but not a group. In fact, a bistochastic matrix with non-negative entries has a bistochastic inverse with non-negative entries if and only if it is a permutation matrix [(:ref:MonPle73)]. | ||
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