Givens rotation
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In mathematics, a Givens rotation is a matrix of the form
where c = cos(θ) and s = sin(θ) appear in the ith / kth row and column, respectively.
The Givens rotation matrix is the identity matrix with these changes:
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; one advantage over Householder transformations is that they can easily be parallelised, and another is that for many very sparse matrices they have lower operation count.
[edit] Example
Example with i = 2 and k = 4.
[edit] References
- Gene H. Golub and Charles F. van Loan, Matrix Computations, 2nd edn., The Johns Hopkins University Press, 1989.
