The Eigenvectors in the Jacobi Method

Eventually we end up with a matrix D, which is diagonal to requested precision. The obtained transformation is:

$$\boldsymbol{D} = \boldsymbol{V}^T\cdot\boldsymbol{A}\cdot\boldsymbol{V}$$ (3.35)

where

$$\boldsymbol{V} = \boldsymbol{P}_1\cdot\boldsymbol{P}_2\cdot\boldsymbol{P}_3 \cdot\ldots\cdot\boldsymbol{P}_n$$ (3.36)

Given matrix V we can compute the new matrix V'as follows:

$$v_{rs} \qquad \hbox{for} \qquad s\neq p, s\neq q$$ v'_rs = (3.37)

$$cv_{rp} - s v_{rq} = v_{rp} - s \left(v_{rq} + \tau v_{rp}\right),$$ v'_rp = (3.38)

$$c v_{rq} + s v_{rp} = v_{rq} + s \left(v_{rp} - \tau v_{rq}\right)$$ v'_rq = (3.39)