Discussion of Jacobi's method for eigenvalues

It means that its matrix elements that differ from zero are given by $$ s_{kk}= s_{ll}=\cos\theta, s_{kl}=-s_{lk}= -\sin\theta, s_{ii}=1\hspace{0.5cm} i\ne k \hspace{0.5cm} i \ne l, $$ A similarity transformation $$ \mathbf{B}= \mathbf{S}^T \mathbf{A}\mathbf{S}, $$ results in $$ \begin{align*} b_{ik} =& a_{ik}\cos\theta - a_{il}\sin\theta , i \ne k, i \ne l \\ b_{il} =& a_{il}\cos\theta + a_{ik}\sin\theta , i \ne k, i \ne l \nonumber\\ b_{kk} =& a_{kk}\cos^2\theta - 2a_{kl}\cos\theta \sin\theta +a_{ll}\sin^2\theta\nonumber\\ b_{ll} =& a_{ll}\cos^2\theta +2a_{kl}\cos\theta sin\theta +a_{kk}\sin^2\theta\nonumber\\ b_{kl} =& (a_{kk}-a_{ll})\cos\theta \sin\theta +a_{kl}(\cos^2\theta-\sin^2\theta)\nonumber \end{align*} $$ The angle \( \theta \) is arbitrary. The recipe is to choose \( \theta \) so that all non-diagonal matrix elements \( b_{kl} \) become zero.