The factor \( e_1 \) is a possibly non-vanishing element. The next transformation produced by \( \mathbf{S}_2 \) has the same effect as \( \mathbf{S}s \) but now on the submatirx \( \mathbf{A^{'}} \) only $$ \left (\mathbf{S}_{1}\mathbf{S}_{2} \right )^{T} \mathbf{A}\mathbf{S}_{1} \mathbf{S}_{2} = \left( \begin{array}{ccccccc} a_{11} & e_1 & 0 & 0 & \dots &0 & 0 \\ e_1 & a'_{22} &e_2 & 0 & \dots &\dots &0 \\ 0 & e_2 &a''_{33} & \dots & \dots &\dots &a''_{3n} \\ 0 & \dots &\dots & \dots & \dots &\dots & \\ 0 & 0 &a''_{n3} & \dots & \dots &\dots &a''_{nn} \\ \end{array} \right) $$ Note that the effective size of the matrix on which we apply the transformation reduces for every new step. In the previous Jacobi method each similarity transformation is in principle performed on the full size of the original matrix.