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,
Read more about Jacobi Rotation: Numerically Stable Computation
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