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where c = cos(θ) and s = sin(θ) appear in the i-th / k-th row and column, respectively. More formally,
The product G(i,k,θ)Tx represents a counter-clockwise rotation of the vector x in the (i,k) plane about θ radians, hence the name Givens rotation.
The main use of Givens rotations in numerical linear algebra is to introduce zeros in vectors/matrices. This effect can e.g. be employed for computing the QR decomposition of a matrix; their advantage over Householder transformations is that they can easily be parallelised.
- Gene H. Golub and Charles F. van Loan, Matrix Computations, 2nd edn., The Johns Hopkins University Press, 1989.
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