# 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. 1.

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

2. 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 matrix rotates a vector (in $\mathbbmss{R}^{2}$) $\theta$ radians in the anti-clockwise direction.

## Properties

1. 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. 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