In numerical linear algebra, a Jacobi rotation is a rotation, Qkℓ, of a 2-dimensional linear subspace of an n-dimensional inner product space, chosen to zero a symmetric pair of off-diagonal entries of an n×n real symmetric matrix, A. It is the core operation in the Jacobi eigenvalue algorithm, which is numerically stable and well-suited to implementation on parallel processors .
Formulation The Jacobi rotation is applied as a similarity transformation:
A ↦ Q k ℓ T A Q k ℓ = A ′ . {\displaystyle A\mapsto Q_{k\ell }^{T}AQ_{k\ell }=A'.\,\!}
[ ∗ ⋯ ∗ ⋱ a k k ⋯ a k ℓ ⋮ ⋮ ⋱ ⋮ ⋮ a ℓ k ⋯ a ℓ ℓ ⋱ ∗ ⋯ ∗ ] → [ ∗ ⋯ ∗ ⋱ a k k ′ ⋯ 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 ⋯ a ℓ ℓ ′ ⋱ ∗ ⋯ ∗ ] . {\displaystyle {\begin{bmatrix}{*}&&&\cdots &&&*\\&\ddots &&&&&\\&&a_{kk}&\cdots &a_{k\ell }&&\\\vdots &&\vdots &\ddots &\vdots &&\vdots \\&&a_{\ell k}&\cdots &a_{\ell \ell }&&\\&&&&&\ddots &\\{*}&&&\cdots &&&*\end{bmatrix}}\to {\begin{bmatrix}{*}&&&\cdots &&&*\\&\ddots &&&&&\\&&a'_{kk}&\cdots &0&&\\\vdots &&\vdots &\ddots &\vdots &&\vdots \\&&0&\cdots &a'_{\ell \ell }&&\\&&&&&\ddots &\\{*}&&&\cdots &&&*\end{bmatrix}}.}
Only rows k and ℓ and columns k and ℓ of A will be affected, and that A′ will remain symmetric. Also, an explicit matrix for Qkℓ is rarely computed; instead, auxiliary values are computed and A is updated in an efficient and numerically stable way. However, for reference, we may write the matrix as
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