rotation matrix

Definition 1.

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

Examples

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

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

   where $\theta \in ℝ$. Multiplication (from the left) with this matrix rotates a vector (in ${ℝ}^2$) $\theta$ radians in the anti-clockwise direction.

Properties

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

2. In fact, for $v\in {ℝ}^n$, $n\ge 3$, there are many rotation matrices $𝐑\in \mathrm{SO}(n)$ such that $R\cdot v={(1,0,\mathrm{\dots },0)}^T$. To see this, let $f$ be the mapping $f:\mathrm{SO}(n-1)\to \mathrm{SO}(n)$, defined as

   Then for each $Q\in \mathrm{SO}(n-1)$, $f(Q)$ maps ${(1,0,\mathrm{\dots },0)}^T$ onto itself. Thus, if $R_0\in \mathrm{SO}(n)$ satisfies $R\cdot v={(1,0,\mathrm{\dots },0)}^T$, then $f(Q)\cdot R$ satisfies the same property for all $Q\in \mathrm{SO}(n-1)$.

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