PlanetMath: rotation matrix

(more info)

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rotation matrix

(Definition)

Definition 1 A rotation matrix is a (real) orthogonal matrix whose determinant is . All $n\times n$ rotation matrices form a group called the special orthogonal group and it is denoted by $\operatorname{SO}(n)$ .

Examples

The identity matrix in $\mathbbmss{R}^n$ is a rotation matrix.
The most general rotation matrix in $\mathbbmss{R}^2$ can be written as
$\displaystyle \begin{pmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{pmatrix},$
where $\theta\in \mathbbmss{R}$ . Multiplication (from the left) with this matrix rotates a vector (in $\mathbbmss{R}^2$ ) $\theta$ radians in the anti-clockwise direction.

Properties

Suppose $v\in \mathbbmss{R}^n$ is a unit vector. Then there exists a rotation matrix such that $R\cdot v = (1,0,\ldots, 0)$ .
In fact, for $v\in \mathbbmss{R}^n$ , $n\ge 3$ , there are many rotation matrices $\mathbf{R} \in \operatorname{SO}(n)$ such that $R\cdot v = (1,0,\ldots, 0)^T$ . To see this, let be the mapping $f\colon \operatorname{SO}(n-1)\rightarrow \operatorname{SO}(n)$ , defined as
$\displaystyle f(Q)= \begin{pmatrix} 1 & 0_{1\times n-1}\ 0_{n-1\times 1} & Q_{n-1\times n-1} \end{pmatrix}.$
Then for each $Q\in \operatorname{SO}(n-1)$ , maps $(1,0,\ldots, 0)^T$ onto itself. Thus, if $R_0 \in \operatorname{SO}(n)$ satisfies $R\cdot v=(1,0,\ldots, 0)^T$ , then $f(Q)\cdot R$ satisfies the same property for all $Q\in \operatorname{SO}(n-1)$ .