The following simple example may help in understanding the meaning of the transition matrix $\hat{W}$ and the vector $\hat{w}$. Consider the $4\times 4$ matrix $\hat{W}$ $$\begin{equation*} \hat{W} = \left(\begin{array}{cccc} 1/4 & 1/9 & 3/8 & 1/3 \\ 2/4 & 2/9 & 0 & 1/3\\ 0 & 1/9 & 3/8 & 0\\ 1/4 & 5/9& 2/8 & 1/3 \end{array} \right), \end{equation*}$$ and we choose our initial state as $$\begin{equation*} \hat{w}(t=0)= \left(\begin{array}{c} 1\\ 0\\ 0 \\ 0 \end{array} \right). \end{equation*}$$