The dynamical equation can be written as
$$ \begin{equation} {\cal w}^{(n)}_i = \sum_j M_{ij}{\cal w}^{(n-1)}_j \tag{10} \end{equation} $$with the matrix \( M \) given by
$$ \begin{equation} M_{ij} = \delta_{ij}\left [ 1 -\sum_k T_{i\rightarrow k} A_{i \rightarrow k} \right ] + T_{j\rightarrow i} A_{j\rightarrow i} \,. \tag{11} \end{equation} $$Summing over \( i \) shows that \( \sum_i M_{ij} = 1 \), and since \( \sum_k T_{i\rightarrow k} = 1 \), and \( A_{i \rightarrow k} \leq 1 \), the elements of the matrix satisfy \( M_{ij} \geq 0 \). The matrix \( M \) is therefore a stochastic matrix.