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Bistochastic group

A bistochastic group is any member of a sequence $(B_N)_{N\in \N}$ of classical matrix groups.

Definition

For given $N\in \N$ and any scalar $N\times N$ -matrix $u=(u_{i,j})_{i,j=1}^N\in \C^{N\times N}$ with $u_{i,j}\geq 0$ for all $i,j\in\{1,\ldots,N\}$ , i.e., with non-negative entries, the matrix $A$ is called

right stochastic if $\sum_{\ell=1}^N u_{i,\ell}=1$ for all $i\in \{1,\ldots,N\}$ , i.e., if each row of $u$ sums up to $1$ ,
left stochastic if $\sum_{k=1}^N u_{k,j}=1$ for all $j\in \{1,\ldots,N\}$ , i.e., if each column of $u$ sums up to $1$ ,
bistochastic or doubly stochastic if $u$ is both right and left stochastic.

Right or left stochastic matrices are also known as probability matrices, transition matrices, substitution matrices or Markov matrices.

The set of bistochastic $N\times N$ -matrices forms a compact Hausdorff semigroup with respect to the topology inherited from $\C^{N\times N}$ . The inverse of a regular bistochastic matrix is generally not bistochastic. In fact, a bistochastic matrix is regular if and only if it is a permutation matrix [MonPle73].

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

$B_N\colon\hspace{-0.6em}=\{ u\in \C^{N\times N}\,\vert\, u=\overline{u},\, uu^t=u^tu=I, \forall_{i,j=1}^N: u_{i,j}\geq 0, \, {\textstyle\sum_{\ell=1}^N} u_{i,\ell}={\textstyle\sum_{k=1}^N} u_{k,j}=1\},$

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.