rotation matrix

Definition 1.

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

Examples

1.

The identity matrix in $\mathbbmss{R}^{n}$ is a rotation matrix.

The most general rotation matrix in $\mathbbmss{R}^{2}$ can be written as

$\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix},$

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

1.

Suppose $v\in\mathbbmss{R}^{n}$ is a unit vector. Then there exists a rotation matrix $R$ such that $R\cdot v=(1,0,\ldots,0)$ .

In fact, for $v\in\mathbbmss{R}^{n}$ , $n\geq 3$ , 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 $f$ be the mapping $f\colon\operatorname{SO}(n-1)\rightarrow\operatorname{SO}(n)$ , defined as

$f(Q)=\begin{pmatrix}1&0_{1\times n-1}\\ 0_{n-1\times 1}&Q_{n-1\times n-1}\end{pmatrix}.$

Then for each $Q\in\operatorname{SO}(n-1)$ , $f(Q)$ 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 $Q\in\operatorname{SO}(n-1)$ .

Title	rotation matrix
Canonical name	RotationMatrix
Date of creation	2013-03-22 15:03:57
Last modified on	2013-03-22 15:03:57
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	17
Author	matte (1858)
Entry type	Definition
Classification	msc 15-00
Synonym	rotational matrix
Related topic	OrthogonalMatrices
Related topic	ExampleOfRotationMatrix
Related topic	DecompositionOfOrthogonalOperatorsAsRotationsAndReflections
Related topic	DerivationOfRotationMatrixUsingPolarCoordinates
Related topic	DerivationOf2DReflectionMatrix
Related topic	TransitionToSkewAngledCoordinates