Jacobi uses a sequence of plane rotations to reduce a symmetric matrix A to a diagonal matrix
\[A_{0} = A, \> A_{k+1} = J_{k}^{T}A_{k}J_{k}, \> A_{k} \rightarrow \Sigma \> as \> k \rightarrow \infty\]
Each plane rotation, \(J_{k} = J_{k}(i, j, \theta)\), now called a Jacobi or Givens rotation
(2)\[\begin{split}\begin{equation}
J_{k}(i, j, \theta)=\begin{vmatrix}
I &\, & & & \\
\, &\,c & &s & \\
\, &\, &I & & \\
\, &-s & &c & \\
& &\, & & &I
\end{vmatrix}
\end{equation}\end{split}\]
where \(c=cos \theta\) and \(s=sin \theta\). The angle \(\theta\) is chosen to eliminate the pair \(a_{ij}\), \(a_{ji}\) by applying \(J(i,j, \theta )\) on the left and right of \(A\), which can be viewed as the 2x2 eigenvalue problem
\[\begin{split}\begin{equation}
\hat{J}_{(k)}^{T} \hat{A}_{(k)} \hat{J}_{(k)}= \begin{vmatrix}
\, c &s \\
-s &c
\end{vmatrix}^{T} \begin{vmatrix}
a_{ii} &a_{ij} \\
a_{ji} &a_{jj}
\end{vmatrix} \begin{vmatrix}
d_{ii} &0 \\
0 &d_{jj}
\end{vmatrix}= \hat{A}_{(k+1)}
\end{equation}\end{split}\]
where \(\hat{A}\) is a 2X2 submatrix of matrix A. After the Givens rotations of the whole matrix A, the off-diagonal value of A will be reduced after 5-10 times iteration of the process.