Editing Givens rotation (section)

== As action on matrices ==
A Givens rotation acting on a matrix from the left is a row operation, moving data between rows but always within the same column. Unlike the [[elementary matrix#Row-addition_transformations|elementary operation of row-addition]], a Givens rotation changes both of the rows addressed by it. To understand how it is a rotation, one may denote the elements of one target row by <math>x_1</math> through <math>x_n</math> and the elements of the other target row by <math>y_1</math> through <math>y_n</math>:
<math display="block">
  \begin{bmatrix}
    \vdots & \vdots & \ddots & \vdots \\
    x_1 & x_2 & \dots & x_n \\
    \vdots & \vdots & \ddots & \vdots \\
    y_1 & y_2 & \dots & y_n \\
    \vdots & \vdots & \ddots & \vdots
  \end{bmatrix}
</math>
Then the effect of a Givens rotation is to rotate each subvector <math>(x_k,y_k)</math> by the same angle. As with row-addition, algorithms often choose this angle so that one specific element becomes zero, and whatever happens in remaining columns is considered acceptable side-effects.

A Givens rotation acting on a matrix from the right is instead a column operation, moving data between two columns but always within the same row. As with action from the left, it rotates each subvector <math>(x_k,y_k)</math> by the same angle, but here these named elements occur in the matrix as
<math display="block">
  \begin{bmatrix}
    \dots & x_1 & \dots & y_1 & \dots \\
    \dots & x_2 & \dots & y_2 & \dots \\
    \ddots & \vdots & \ddots & \vdots & \ddots \\
    \dots & x_n & \dots & y_n & \dots 
  \end{bmatrix}
</math>
Some algorithms, especially those concerned with preserving [[matrix similarity]], apply Givens rotations as a [[Conjugacy_class#Conjugacy_as_group_action|conjugate action]]: both rotating by one angle between two rows, and rotating by the same angle between the corresponding columns. In this case the effect on the four elements affected by both rotations is more complicated; a [[Jacobi rotation]] is such a conjugate action to the end of zeroing the two off-diagonal elements among these four.

The main use of Givens rotations in [[numerical linear algebra]] is to transform vectors or matrices into a special form with zeros in certain coefficients. This effect can, for example, be employed for computing the [[QR decomposition]] of a matrix. One advantage over [[Householder transformation]]s is that they can easily be parallelised, and another is that often for very sparse matrices they have a lower operation count.