It remains to find a recipe for determining the transformation \( \mathbf{S}_n \). We illustrate the method for \( \mathbf{S}_1 \) which we assume takes the form $$ \mathbf{S_{1}} = \left( \begin{array}{cc} 1 & \mathbf{0^{T}} \\ \mathbf{0}& \mathbf{P} \end{array} \right), $$ with \( \mathbf{0^{T}} \) being a zero row vector, \( \mathbf{0^{T}} = \{0,0,\cdots\} \) of dimension \( (n-1) \). The matrix \( \mathbf{P} \) is symmetric with dimension (\( (n-1) \times (n-1) \)) satisfying \( \mathbf{P}^2=\mathbf{I} \) and \( \mathbf{P}^T=\mathbf{P} \). A possible choice which fullfils the latter two requirements is $$ \mathbf{P}=\mathbf{I}-2\mathbf{u}\mathbf{u}^T, $$ where \( \mathbf{I} \) is the \( (n-1) \) unity matrix and \( \mathbf{u} \) is an \( n-1 \) column vector with norm \( \mathbf{u}^T\mathbf{u} \) (inner product).