Givens rotation
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In mathematics, a Givens rotation is a matrix of the form
- <math>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}<math>
where c = cos(θ) and s = sin(θ) appear in the i-th / k-th row and column, respectively. More formally,
- <math>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}.<math>
The product <math>G(i, k, \theta)^Tx<math> 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.