The matrix \( \hat{U} \) takes then the form $$ \hat{U} = \left( \begin{array}{ccccccccc} x&x &x &0 &0 & \dots &0 & 0 & 0\\ 0&x &x &x &0 & \dots &0 & 0 & 0\\ 0&0 &x &x &x & \dots &0 & 0 & 0\\ \dots&\dots &\dots &\dots &\dots &\dots \\ 0&0 &0 & 0 & 0 & \dots &x &x &x \\ 0&0 &0 & 0 & 0 & \dots &0 &x &x \\ 0&0 &0 & 0 & 0 & \dots &0 &0 & x \end{array} \right) $$ which has a second superdiagonal.