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rotation matrix
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(Definition)
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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 $\operatorname{SO}(n)$ .
- The identity matrix in
 is a rotation matrix.
- The most general rotation matrix in
 can be written as $$ \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}, $$ where
 . Multiplication (from the left) with this matrix rotates a vector (in
 ) $\theta$ radians in the anti-clockwise direction.
- Suppose
 is a unit vector. Then there exists a rotation matrix $R$ such that $R\cdot v = (1,0,\ldots, 0)$ .
- In fact, for
 , $n\ge 3$ , 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 $f$ 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 $Q\in \operatorname{SO}(n-1)$ .
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"rotation matrix" is owned by matte. [ full author list (3) ]
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Cross-references: property, onto, mapping, unit vector, radians, vector, rotates, matrix, multiplication, identity matrix, group, determinant, orthogonal matrix, real
There are 14 references to this entry.
This is version 14 of rotation matrix, born on 2005-02-19, modified 2006-06-13.
Object id is 6786, canonical name is RotationMatrix.
Accessed 21936 times total.
Classification:
| AMS MSC: | 15-00 (Linear and multilinear algebra; matrix theory :: General reference works ) |
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Pending Errata and Addenda
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