In order to determine each $\mathbf{S}_i$ let us see what happens after the first multiplication, namely, $$\mathbf{S}_1^T\mathbf{A}\mathbf{S}_1= \left( \begin{array}{ccccccc} a_{11} & e_1 & 0 & 0 & \dots &0 & 0 \\ e_1 & a'_{22} &a'_{23} & \dots & \dots &\dots &a'_{2n} \\ 0 & a'_{32} &a'_{33} & \dots & \dots &\dots &a'_{3n} \\ 0 & \dots &\dots & \dots & \dots &\dots & \\ 0 & a'_{n2} &a'_{n3} & \dots & \dots &\dots &a'_{nn} \\ \end{array} \right)$$ where the primed quantities represent a matrix $\mathbf{A}'$ of dimension $n-1$ which will subsequentely be transformed by $\mathbf{S}_2$.