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 calledthe special orthogonal group and it is denoted by $\\operatorname{SO}(n)$ .

Examples

1.
The identity matrix in $\\mathbbmss{R}^{n}$ is a rotation matrix.
2.
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 matrixrotates a vector (in $\\mathbbmss{R}^{2}$ ) $\\theta$ radians in the anti-clockwisedirection.

Properties

1.
Suppose $v\\in\\mathbbmss{R}^{n}$ is a unit vector.Then there exists a rotation matrix $R$ such that $R\\cdot v=(1,0,\\ldots,0)$ .
2.
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