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, when applied as a similarity transformation: It is the core operation in the Jacobi eigenvalue algorithm, which is numerically stable and well-suited to implementation on parallel processors. Suppose h is an index other than k or ℓ (which must themselves be distinct). Then the similarity update produces, algebraically,
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| - Jacobi rotation (en)
- Rotação de Jacobi (pt)
- 雅可比旋转 (zh)
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| - Em álgebra linear numérica, uma rotação de Jacobi é uma rotação, Qkℓ, de um subespaço bi-dimensional de um espaço n-dimensional com espaço com produto interno, escolhida de modo que sejam zerados dois elementos simétricos não pertencentes à diagonal principal de uma matriz n×n simétrica e real, A, quando aplicada como uma transformação de similaridade: Tais rotações são a operação principal no algoritmo de autovalores de Jacobi, que é e adequado para a implementação em processadores paralelos. (pt)
- 在数值线性代数中，雅可比旋转是 n 维内积空间的二维线性子空间的旋转 Qkℓ，在用做的时候，被选择来置零 n×n 实数对称矩阵 A 的非对角元素的对称对: 它是的核心运算，它是数值上稳定的并适合用并行计算实现。 注意到只有 A 的行 k 和 ℓ 与列 k 和 ℓ 受到影响，并且 A′ 将保持对称。还有给 Qkℓ 的明显的矩阵很少被计算，转而计算辅助值，A 也有效率和数值上稳定的方式更新。但是，为了引用，我们写矩阵为 就是说，除了四个元素之外，Qkℓ 是一个单位矩阵，两个在对角线上(qkk 和 qℓℓ 都等于 c) 而两个位于远离对角的位置上(qkℓ 和 Qℓk 分别等于 s 和 −s)。这里的 c = cos ϑ 而 s = sin ϑ 对于某个角度 ϑ；但是对于应用这种旋转，这个角度自身是不需要的。使用克罗内克δ符号，矩阵元素可以写为 假设 h 是不为 k 或 ℓ 的索引(它们自身必须是不同的)。类似的更改过程在代数上写为 (zh)
- 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, when applied as a similarity transformation: It is the core operation in the Jacobi eigenvalue algorithm, which is numerically stable and well-suited to implementation on parallel processors. Suppose h is an index other than k or ℓ (which must themselves be distinct). Then the similarity update produces, algebraically, (en)
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| - 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, when applied as a similarity transformation: It is the core operation in the Jacobi eigenvalue algorithm, which is numerically stable and well-suited to implementation on parallel processors. 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 That is, Qkℓ is an identity matrix except for four entries, two on the diagonal (qkk and qℓℓ, both equal to c) and two symmetrically placed off the diagonal (qkℓ and qℓk, equal to s and −s, respectively). Here c = cos θ and s = sin θ for some angle θ; but to apply the rotation, the angle itself is not required. Using Kronecker delta notation, the matrix entries can be written Suppose h is an index other than k or ℓ (which must themselves be distinct). Then the similarity update produces, algebraically, (en)
- Em álgebra linear numérica, uma rotação de Jacobi é uma rotação, Qkℓ, de um subespaço bi-dimensional de um espaço n-dimensional com espaço com produto interno, escolhida de modo que sejam zerados dois elementos simétricos não pertencentes à diagonal principal de uma matriz n×n simétrica e real, A, quando aplicada como uma transformação de similaridade: Tais rotações são a operação principal no algoritmo de autovalores de Jacobi, que é e adequado para a implementação em processadores paralelos. (pt)
- 在数值线性代数中，雅可比旋转是 n 维内积空间的二维线性子空间的旋转 Qkℓ，在用做的时候，被选择来置零 n×n 实数对称矩阵 A 的非对角元素的对称对: 它是的核心运算，它是数值上稳定的并适合用并行计算实现。 注意到只有 A 的行 k 和 ℓ 与列 k 和 ℓ 受到影响，并且 A′ 将保持对称。还有给 Qkℓ 的明显的矩阵很少被计算，转而计算辅助值，A 也有效率和数值上稳定的方式更新。但是，为了引用，我们写矩阵为 就是说，除了四个元素之外，Qkℓ 是一个单位矩阵，两个在对角线上(qkk 和 qℓℓ 都等于 c) 而两个位于远离对角的位置上(qkℓ 和 Qℓk 分别等于 s 和 −s)。这里的 c = cos ϑ 而 s = sin ϑ 对于某个角度 ϑ；但是对于应用这种旋转，这个角度自身是不需要的。使用克罗内克δ符号，矩阵元素可以写为 假设 h 是不为 k 或 ℓ 的索引(它们自身必须是不同的)。类似的更改过程在代数上写为 (zh)
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