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Metropolis algorithm, setting it up

Since the probability of making some transition must be 1, \sum_j T_{i\rightarrow j} = 1 , and Eq. (5) becomes

\begin{equation} {\cal w}^{(n)}_i = {\cal w}^{(n-1)}_i + \sum_j \left [ {\cal w}^{(n-1)}_jT_{j\rightarrow i} A_{j\rightarrow i} -{\cal w}^{(n-1)}_iT_{i\rightarrow j}A_{i\rightarrow j} \right ] \,. \tag{6} \end{equation}