We note that both the vector and the matrix are properly normalized. Summing the vector elements gives one and summing over columns for the matrix results also in one. Furthermore, the largest eigenvalue is one. We act then on \( \hat{w} \) with \( \hat{W} \). The first iteration is $$ \begin{equation*} \hat{w}(t=\epsilon) = \hat{W}\hat{w}(t=0), \end{equation*} $$

resulting in $$ \begin{equation*} \hat{w}(t=\epsilon)= \left(\begin{array}{c} 1/4\\ 1/2 \\ 0 \\ 1/4 \end{array} \right). \end{equation*} $$