Givens rotation

Builds on Rotation Orthogonal matrix Identity matrix

Definition

Let $n \ge 2$ , let $1 \le i < j \le n$ , and let $c, s$ be real numbers with $c^2 + s^2 = 1$ . The Givens rotation $G(i, j; c, s)$ is the $n \times n$ real matrix that agrees with the identity matrix except in the four entries $g_{ii} = c, \qquad g_{ij} = -s, \qquad g_{ji} = s, \qquad g_{jj} = c.$ Writing $c = \cos\theta$ and $s = \sin\theta$ , it is the rotation by the angle $\theta$ in the plane of the coordinates $i$ and $j$ .

Remarks

$G = G(i, j; c, s)$ is an orthogonal matrix with determinant $c^2 + s^2 = 1$ . For $x \in \R^n$ , the product $G x$ changes only the entries $i$ and $j$ : $(Gx)_i = c x_i - s x_j$ and $(Gx)_j = s x_i + c x_j$ . If $(x_i, x_j) \ne (0, 0)$ , put $r = \sqrt{x_i^2 + x_j^2}$ , $c = x_i / r$ and $s = -x_j / r$ ; then $(Gx)_i = r$ and $(Gx)_j = 0$ . Example: for $x = (3, 4)$ in $\R^2$ , $r = 5$ , $c = 3/5$ , $s = -4/5$ , and $G x = (9/5 + 16/5, \ -12/5 + 12/5) = (5, 0)$ . Non-example: the matrix with $c = s = 1$ in these four places is not a Givens rotation, since $c^2 + s^2 = 2$ , and it is not orthogonal. Other books put $-s$ in position $(j, i)$ instead; the two conventions differ by the sign of $\theta$ .

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Euclidean isometry concept
Euclidean plane object
Identity matrix object
Matrix definition
Rotation definition
Transpose definition
Orthogonal matrix definition
Givens rotation definition

Tag	`X600GRHDQ7`
URI	`m2://X600GRHDQ7`
Kind	definition
Writing	W0 stub
MSC	65F25, 15A23
Defines	Givens rotation
Uses	Rotation, Orthogonal matrix, Identity matrix

Givens rotation

Definition

Remarks

Reading path

Further reading

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