The resulting matrix reads $$\mathbf{S}^{T} \mathbf{A} \mathbf{S} = \left( \begin{array}{ccccccc} a_{11} & e_1 & 0 & 0 & \dots &0 & 0 \\ e_1 & a'_{22} & e_2 & 0 & \dots &0 &0 \\ 0 & e_2 & a''_{33} & e_3 &0 &\dots & 0\\ \dots & \dots & \dots & \dots &\dots &\dots & \dots\\ 0 & \dots & \dots & \dots &\dots &a^{(n-1)}_{n-2} & e_{n-1}\\ 0 & \dots & \dots & \dots &\dots &e_{n-1} & a^{(n-1)}_{nn} \end{array} \right) .$$