Convergence of the Jacobi Method

Consider

$$S = \sum_{r \neq s} \vert a_{rs} \vert^2$$ (3.34)

The transformation $S \rightarrow S'$ is orthogonal and it results in increase in the sum of the squares of the diagonal elements by $2 t^2 \vert a_{pq}\vert^2$. Consequently, the sum of the squares of the off-diagonal terms must decrease by $2 t^2 \vert a_{pq}\vert^2$. Because S is limited from below, it must be always greater than or equal to zero, and it diminishes with every step, eventually we must converge.