Consider an example of an (\( n\times n \)) orthogonal transformation matrix $$ \mathbf{S}= \left( \begin{array}{cccccccc} 1 & 0 & \dots & 0 & 0 & \dots & 0 & 0 \\ 0 & 1 & \dots & 0 & 0 & \dots & 0 & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots & 0 & \dots \\ 0 & 0 & \dots & \cos\theta & 0 & \dots & 0 & \sin\theta \\ 0 & 0 & \dots & 0 & 1 & \dots & 0 & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots & 1 & \dots \\ 0 & 0 & \dots & -\sin\theta & 0 & \dots & 0 & \cos\theta \end{array} \right) $$ with property \( \mathbf{S^{T}} = \mathbf{S^{-1}} \). It performs a plane rotation around an angle \( \theta \) in the Euclidean $n-$dimensional space.