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== Description ==
Let ''AS'' be a symmetric matrix, and ''G'' = ''G''(''i'',''j'',''θ'') be a [[Givens rotation|Givens rotation matrix]]. Then:
:<math>AS'=G^\top AS G \, </math>
is symmetric and [[similar matrix|similar]] to ''AS''.
Furthermore, ''AS′'' has entries:
:<math>\begin{align}
AS'_{ii} &= c^2\, A_S_{ii} - 2\, s c \,A_S_{ij} + s^2\, A_S_{jj} \\
AS'_{jj} &= s^2 \,A_S_{ii} + 2 s c\, A_S_{ij} + c^2 \, A_S_{jj} \\
AS'_{ij} &= AS'_{ji} = (c^2 - s^2 ) \, A_S_{ij} + s c \, (A_S_{ii} - A_S_{jj} ) \\
AS'_{ik} &= AS'_{ki} = c \, A_S_{ik} - s \, A_S_{jk} & k \ne i,j \\
AS'_{jk} &= AS'_{kj} = s \, A_S_{ik} + c \, A_S_{jk} & k \ne i,j \\
AS'_{kl} &= A_S_{kl} &k,l \ne i,j
\end{align}</math>
where ''s'' = sin(''θ'') and ''c'' = cos(''θ'').
Since ''G'' is orthogonal, ''AS'' and ''AS''′ have the same [[Frobenius norm]] ||·||<sub>F</sub> (the square-root sum of squares of all components), however we can choose ''θ'' such that ''AS''′<sub>''ij''</sub> = 0, in which case ''AS''′ has a larger sum of squares on the diagonal:
:<math> AS'_{ij} = \cos(2\theta) A_S_{ij} + \tfrac{1}{2} \sin(2\theta) (A_S_{ii} - A_S_{jj}) </math>
Set this equal to 0, and rearrange:
:<math> \tan(2\theta) = \frac{2 A_S_{ij}}{A_S_{jj} - A_S_{ii}} </math>
if <math> A_S_{jj} = A_S_{ii} </math>
:<math> \theta = \frac{\pi} {4} </math>
In order to optimize this effect, ''AS''<sub>''ij''</sub> should be the largest off-diagonal component, called the ''pivot''.
The Jacobi eigenvalue method repeatedly performs rotations until the matrix becomes almost diagonal. Then the elements in the diagonal are approximations of the (real) eigenvalues of ''AS''.
== Convergence ==
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