The main idea is thus to reduce systematically the norm of the off-diagonal matrix elements of a matrix $\mathbf{A}$ $$\mathrm{off}(\mathbf{A}) = \sqrt{\sum_{i=1}^n\sum_{j=1,j\ne i}^n a_{ij}^2}.$$ To demonstrate the algorithm, we consider the simple $2\times 2$ similarity transformation of the full matrix. The matrix is symmetric, we single out $ 1 \le k < l \le n$ and use the abbreviations $c=\cos\theta$ and $s=\sin\theta$ to obtain $$\left( \begin{array}{cc} b_{kk} & 0 \\ 0 & b_{ll} \\\end{array} \right) = \left( \begin{array}{cc} c & -s \\ s &c \\\end{array} \right) \left( \begin{array}{cc} a_{kk} & a_{kl} \\ a_{lk} &a_{ll} \\\end{array} \right) \left( \begin{array}{cc} c & s \\ -s & c \\\end{array} \right).$$