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).