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Givens rotation

In mathematics, a Givens rotation is a matrix of the form

$G(i, k, \theta) = \begin{bmatrix} 1 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\ \vdots & \ddots & \vdots & & \vdots & & \vdots \\ 0 & \cdots & c & \cdots & s & \cdots & 0 \\ \vdots & & \vdots & \ddots & \vdots & & \vdots \\ 0 & \cdots & -s & \cdots & c & \cdots & 0 \\ \vdots & & \vdots & & \vdots & \ddots & \vdots \\ 0 & \cdots & 0 & \cdots & 0 & \cdots & 1 \end{bmatrix}$

where c = cos(θ) and s = sin(θ) appear in the i-th / k-th row and column, respectively. More formally,

$G(i, k, \theta)_{j, l} = \begin{cases} \cos\theta & \mbox{ if } j = i, l = i \mbox{ or } j = k, l = k \\ \sin\theta & \mbox{ if } j = i, l = k \\ -\sin\theta & \mbox{ if } j = k, l = i \\ 1 & \mbox{ if } j = l \\ 0 & \mbox{ otherwise } \end{cases}.$

The product $G (i, k,θ) T x$ 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.

References

Gene H. Golub and Charles F. van Loan, Matrix Computations, 2nd edn., The Johns Hopkins University Press, 1989.

Categories: Numerical analysis

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03-10-2013 05:06:04
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