# All Science Fair Projects

## Science Fair Project Encyclopedia for Schools!

 Search    Browse    Forum  Coach    Links    Editor    Help    Tell-a-Friend    Encyclopedia    Dictionary

# Science Fair Project Encyclopedia

For information on any area of science that interests you,
enter a keyword (eg. scientific method, molecule, cloud, carbohydrate etc.).
Or else, you can start by choosing any of the categories below.

# 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)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.

## References

• Gene H. Golub and Charles F. van Loan, Matrix Computations, 2nd edn., The Johns Hopkins University Press, 1989.
03-10-2013 05:06:04
Science kits, science lessons, science toys, maths toys, hobby kits, science games and books - these are some of many products that can help give your kid an edge in their science fair projects, and develop a tremendous interest in the study of science. When shopping for a science kit or other supplies, make sure that you carefully review the features and quality of the products. Compare prices by going to several online stores. Read product reviews online or refer to magazines.

Start by looking for your science kit review or science toy review. Compare prices but remember, Price \$ is not everything. Quality does matter.
 Science Fair Coach What do science fair judges look out for? ScienceHound Science Fair Projects for students of all ages
 All Science Fair Projects.com Site All Science Fair Projects Homepage Search | Browse | Links | From-our-Editor | Books | Help | Contact | Privacy | Disclaimer | Copyright Notice