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Encyclopedia > Rotation matrix

A rotation matrix is a matrix which when multiplied by a vector has the effect of changing the direction of the vector but not its magnitude. Rotation matrices do not include inversions, which can change a right-handed coordinate system into a left-handed coordinate system and vice versa. The set of all rotation matrices along with inversions forms the set of orthogonal matrices. In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... This article gives an overview of the various ways to perform matrix multiplication. ... A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... In geometry, an inversion is a transformation that maps all circles into circles, where by a circle one may also mean a line (a circle with infinite radius). ... In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ...

1 Properties
2 Two dimensions
3 Three dimensions
4 See also
5 External links

Properties

Let $mathcal{M}$ be a general rotation matrix of any dimension: $mathcal{M}inmathbb{R}^{n times n}$

The dot product of two vectors remains unchanged when both are operated upon by a rotation matrix:

$mathbf{a}cdotmathbf{b} = mathcal{M}mathbf{a}cdotmathcal{M}mathbf{b}$

It follows that the inverse of a rotation matrix is its transpose:

$mathcal{M},mathcal{M}^{-1}=mathcal{M},mathcal{M}^top=mathcal{I}$ where $mathcal{I}$ is the identity matrix.

A matrix is a rotation matrix if and only if it is orthogonal and its determinant is unity. The determinant of an orthogonal matrix is ±1; if the determinant is −1, then the matrix also contains a reflection and is not a rotation matrix.

A rotation matrix is orthogonal if its column vectors form an orthonormal basis of $mathbb{R}^{n}$ , that is, the scalar product between any two different column vectors is zero (orthogonality) and the magnitude of each column vector is unity (normalization).

Any rotation matrix can be represented as the exponential of a skew-symmetric matrix A:

$mathcal{M}=exp (mathbf{A})=sum_{k=0}^infty frac{mathbf{A}^k}{k!}$

where the exponential is defined in terms of its Taylor series and $mathbf{A}^k$ is defined in terms of matrix multiplication. The A matrix is known as the generator of the rotation. The Lie algebra of rotation matrices is the algebra of its generators, which is just the algebra of skew-symmetric matrices. The generator can be found by finding the matrix logarithm of M.

It has been suggested that this article or section be merged with invertible matrix. ... In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or Aâ€²) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A... In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... Look up reflection in Wiktionary, the free dictionary. ... In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ... In mathematics, an orthonormal basis of an inner product space V(i. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... In mathematics, a unit vector in a normed vector space is a vector (most commonly a spatial vector) whose length is 1. ... In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji for all i and j. ... As the degree of the Taylor series rises, it approaches the correct function. ... This article gives an overview of the various ways to perform matrix multiplication. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, the logarithm of a matrix is a generalization of the scalar logarithm to matrices. ...

Two dimensions

In two dimensions, a rotation can be defined by a single angle, $θ$ . Conventionally, positive angles represent counter-clockwise rotation. The matrix to rotate a column vector in cartesian coordinates about the origin by a counter-clockwise angle of $θ$ is: In linear algebra, a column vector is an m Ã— 1 matrix, i. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

$M(theta) = begin{bmatrix} cos{theta} & -sin{theta} sin{theta} & cos{theta} end{bmatrix} = expleft(begin{bmatrix} 0 & -theta theta & 0 end{bmatrix}right)$

Three dimensions

In three dimensions, a rotation matrix has one real eigenvalue, equal to unity. The rotation matrix specifies a rotation about the corresponding eigenvector (Euler's rotation theorem). If the angle of rotation is θ then the other two (complex) eigenvalues of the rotation matrix are exp(iθ) and exp(-iθ). It follows that the trace of a 3D rotation matrix is equal to 1 + 2 cos(θ), which can be used to quickly calculate the rotation angle of any 3D rotation matrix. In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics, Eulers rotation theorem states that any rotation has an axis. ... Look up Trace in Wiktionary, the free dictionary. ...

The generators of 3D rotation matrices are 3D skew symmetric matrices. Since only three real numbers are needed to specify a 3D skew-symmetric matrix, it follows that only three real numbers are needed to specify a 3D rotation matrix.

Roll, Pitch and Yaw

A simple way to generate a rotation matrix is to compose it as a sequence of three basic rotations. Rotations about the right-handed cartesian x-, y- and z-axes are known as roll, pitch and yaw rotations respectively. Since these rotations are expressed as a rotation about an axis, their generators are easily expressed.

Rotation around the x-axis is defined as:

$mathcal{R}_x(theta_x)= begin{bmatrix} 1 & 0 & 0 0 & cos{theta_x} & - sin{theta_x} 0 & sin{theta_x} & cos{theta_x} end{bmatrix} =exp left( begin{bmatrix} 0 & 0 & 0 0 & 0 & -theta_x 0 & theta_x & 0 end{bmatrix}right)$ where

θ x

is the roll angle.

Rotation around the y-axis is defined as:

$mathcal{R}_y(theta_y)= begin{bmatrix} cos{theta_y} & 0 & sin{theta_y} 0 & 1 & 0 - sin{theta_y} & 0 & cos{theta_y} end{bmatrix} =expleft( begin{bmatrix} 0 & 0 & theta_y 0 & 0 & 0 - theta_y & 0 & 0 end{bmatrix}right)$ where

θ y

is the pitch angle.

Rotation around the z-axis is defined as:

$mathcal{R}_z(theta_z)= begin{bmatrix} cos{theta_z} & - sin{theta_z} & 0 sin{theta_z} & cos{theta_z} & 0 0 & 0 & 1 end{bmatrix} =expleft( begin{bmatrix} 0 & - theta_z & 0 theta_z & 0 & 0 0 & 0 & 0 end{bmatrix}right)$ where

θ z

is the yaw angle.

In flight dynamics, the roll, pitch and yaw angles are usually given the symbols $γ$ , $α$ , and $β$ , respectively; here, however, the symbols $θ x$ , $θ y$ , and $θ z$ are used to avoid confusion with the Euler angles. The pitch angle of a charged particle is the angle between the particles parrallel motion the local magnetic field. ... The yaw angle is the angle between a vehicles heading and a reference heading (normally true or magnetic North). ... Flight dynamics is the study of orientation of air and space vehicles and how to control the critical flight parameters, typically named pitch, roll and yaw. ... Euler angles are a means of representing the spatial orientation of an object. ...

Any 3-dimensional rotation matrix $mathcal{M}inmathbb{R}^{3times 3}$ can be characterised by the three angles $θ x$ , $θ y$ , and $θ z$ , and may be expressed as a product of the roll, pitch and yaw matrices.

$mathcal{M}$ is a rotation matrix in $mathbb{R}^{3times 3},Leftrightarrow,exist,theta_x,theta_y,theta_zin[0ldotspi):, mathcal{M}=mathcal{R}_z(theta_z),mathcal{R}_y(theta_y),mathcal{R}_x(theta_x)$

The set of all rotations in $mathbb{R}^3$ , together with the operation of composition, form the rotation group SO(3). The matrices discussed here then provide a representation of the group. In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... In mechanics and geometry, the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space, R3. ... Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...

Angle-Axis representation and quaternion representation

Main articles: axis angle and Quaternions and spatial rotation

In three dimensions, a rotation can be defined by a single angle of rotation, $θ$ , and the direction of a unit vector, $hat{mathbf{v}} = (x,y,z)$ , about which to rotate. The axis angle representation of a rotation parameterizes a rotation by two values: an axis, or a line, and an angle describing the magnitude of the rotation about the axis. ... Quaternions provide a convenient mathematical notation for representing orientations and rotations of objects. ... In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ...

$mathcal{M}(hat{mathbf{v}},theta) = begin{bmatrix} cos theta + (1 - cos theta) x^2 & (1 - cos theta) x y - (sin theta) z & (1 - cos theta) x z + (sin theta) y (1 - cos theta) y x + (sin theta) z & cos theta + (1 - cos theta) y^2 & (1 - cos theta) y z - (sin theta) x (1 - cos theta) z x - (sin theta) y & (1 - cos theta) z y + (sin theta) x & cos theta + (1 - cos theta) z^2 end{bmatrix}$

This rotation may be simply expressed in terms of its generator:

$mathcal{M}(hat{mathbf{v}},theta) = expleft( begin{bmatrix} 0 & -ztheta & ytheta ztheta & 0 & -xtheta -ytheta & xtheta & 0 end{bmatrix}right)$

When operating on a vector r, this is equivalent to the Rodrigues' rotation formula In geometry, Rodrigues rotation formula is a vector formula for a rotation in space, given its axis. ...

$mathcal{M} cdot mathbf{r} = mathbf{r} ,cos(theta)+hat{mathbf{v}}times mathbf{r}, sin(theta)+(hat{mathbf{v}}cdotmathbf{r})hat{mathbf{v}}(1-cos(theta))$

The angle-axis representation is closely related to the quaternion representation. In terms of the axis and angle, the quaternion representation is given by a normalized quaternion Q: Quaternions provide a convenient mathematical notation for representing orientations and rotations of objects. ...

$Q=(xi+yj+zk)sin(theta/2)+cos(theta/2),$

where i, j, and k are the three imaginary parts of Q.

Angle-Axis representation via Rotation Tensor

A rotation matrix is not invariant with respect to current reference frame, where the actual rotation is considered. The same physical rotation will have different "rotation matrices" with respect to different sets of basis vectors (orthonormal or not). A rotation tensor is a more general representation of a rotation in space. The representation of rotation by rotation tensors is invariant with respect to change of current reference frame. Each "rotation matrix" representation then is just an "image" of corresponding rotation tensor in a given reference frame. Rotation tensors are constructed using vector dyadics (or "ordered of pairs of vectors"). Dyadics themself can be described as matrices in each given reference frame but are actually much more general objects and are also invariant with respect to rotations of the current reference frame. A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. ...

A rotation tensor $mathcal{M}(hat{mathbf{v}},theta)$ representing a rotation about unit axis $hat{mathbf{v}}$ for angle $θ$ is given by:

$mathcal{M}(hat{mathbf{v}},theta) = hat{mathbf{v}} otimes hat{mathbf{v}} + costheta , ( mathbf{E} - hat{mathbf{v}} otimes hat{mathbf{v}} ) + sintheta , hat{mathbf{v}} times mathbf{E} , ,$

where $mathbf{E}$ is a unit tensor of second order, which is a sum of three dyadics $hat{mathbf{e}}_i otimes hat{mathbf{e}}_i , , , i=1,2,3$ , where $hat{mathbf{e}}_i , , , i=1,2,3$ are three orthogonal unit vectors of any orthonormal reference frame. The given representation does not depend on the actual current orientation of reference frame $hat{mathbf{e}}_i , , , i=1,2,3$ because the unit tensor $mathbf{E}$ itself has the same representation in any orthonormal reference frame (non-orthonormal reference frames will be considered just few lines later).

The Rodrigues' rotation formula simply follows from the above representation as soon as $mathbf{E} cdot mathbf{r} = mathbf{r}$ In geometry, Rodrigues rotation formula is a vector formula for a rotation in space, given its axis. ...

The parts of the the expression for the rotation tensor are easily recognizable.

The dyad $hat{mathbf{v}} otimes hat{mathbf{v}}$ is responsible for a component $(hat{mathbf{v}}cdotmathbf{r})hat{mathbf{v}}$ of the vector $mathbf{R}=mathcal{M} cdot mathbf{r}$ , which is parallel to the axis of rotation $hat{mathbf{v}}$ and is not affected by the multiplication $mathcal{M} cdot mathbf{r}$ . The length of this component is $r , cos alpha$ , where $r$ is the length of the vector $mathbf{r}$ and $α$ is the angle between vectors $hat{mathbf{v}}$ and $mathbf{r}$ .

The projector $(mathbf{E}-hat{mathbf{v}}otimeshat{mathbf{v}})$ gives us a component of the vector $mathbf{r}$ , which is exactly orthogonal to $hat{mathbf{v}}$ . The length of this component is $r , sin alpha$ . This component is then scaled by $cosθ$ depending on the actual rotation angle $θ$ .

And the last part $sin theta , hat{mathbf{v}} times mathbf{E}$ of the expression for the rotation tensor is responsible for a component of the final vector $mathbf{R}$ , which is orthogonal to both $hat{mathbf{v}}$ and $mathbf{r}$ as soon as $(hat{mathbf{v}} times mathbf{E}) cdot mathbf{r} = hat{mathbf{v}} times mathbf{r}$ . The length of vector $(hat{mathbf{v}}timesmathbf{E})cdotmathbf{r}=hat{mathbf{v}}timesmathbf{r}$ is also equal $r , sin alpha$ due to definition of the cross-product of two vectors. For the crossed product in algebra and functional analysis, see crossed product. ...

As a result the three parts $hat{mathbf{v}} otimes hat{mathbf{v}}$ , $(mathbf{E}-hat{mathbf{v}}otimeshat{mathbf{v}})$ and $hat{mathbf{v}}timesmathbf{E}$ of the rotation tensot construct a local orthogonal reference frame which is most convinient for description of the actual rotation of any given vector $mathbf{r}$ .

The above representation can be is generalized onto the case of non-orthonormal reference frame by constructing the unit tensor $mathbf{E}$ as $mathbf{E} = hat{mathbf{e}}_i otimes hat{mathbf{e}}^i$ (assuming Einstein summation), where $hat{mathbf{e}}^i , , , i=1,2,3$ are covectors of vectors $hat{mathbf{e}}_i , , , i=1,2,3$ . A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. ...

The covectors $hat{mathbf{e}}^i$ are build out of $hat{mathbf{e}}_i$ as:

$hat{mathbf{e}}^i = frac{hat{mathbf{e}}_j times hat{mathbf{e}}_k} {hat{mathbf{e}}_i cdot hat{mathbf{e}}_j times hat{mathbf{e}}_k }, ,$

where each triplet ${i, j, k}$ is a cyclic permutations of ${1,2,3}$ triplet. In orthonormal reference frames the vectors $hat{mathbf{e}}_i$ coincide with their "co"-counterparts $hat{mathbf{e}}^i$ .

As a result the given description of rotation in 3D space by the rotation tensor is invariant with respect to any (orthonormal or not) reference frame. Any "rotation matrix" representation is an "image" of the rotation tensor taken in corresponding reference frame.

Euler Angle representation

In three dimensions, a rotation can be defined by three Euler angles, $(α,β,γ)$ . There are a number of possible definitions of the Euler angles. Each may be expressed in terms of a composition of the roll, pitch, and yaw rotations. The rotation matrix expressed in terms of the "z-x-z" Euler angles, in right-handed cartesian coordinates may be expressed as: Euler angles are a means of representing the spatial orientation of an object. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

$mathcal{M}(alpha,beta,gamma)=mathcal{R}_z(alpha)mathcal{R}_x(beta) mathcal{R}_z(gamma)$

carrying out the multiplications yields:

$mathcal{M}(alpha,beta,gamma) = begin{bmatrix} cosalpha cosgamma - cosbeta sinalpha singamma & -cosbeta cosgamma sinalpha - cosalpha singamma & sinalpha sinbeta cosgamma sinalpha + cosalpha cosbeta singamma & cosalpha cosbeta cosgamma - sinalpha singamma & -cosalpha sinbeta sinbeta singamma & cosgamma sinbeta & cosbeta end{bmatrix}$

Since this rotation matrix is not expressed as a rotation about a single axis, its generator is not as simply expressed as in the above examples.

Symmetry Preserving SVD representation

For an axis of rotation $q$ and angle of rotation $θ$ , the rotation matrix

$mathcal{M} = qq^T+QGQ^T$

where the columns of $Q=begin{bmatrix}q_1, & q_2end{bmatrix}$ span the space orthogonal to $q$ and $G$ is the Givens rotation of $θ$ degrees, i.e.

$G = begin{bmatrix} costheta & sintheta -sintheta & costheta end{bmatrix}$

External links

Rotation matrices at Mathworld

Categories: Rotational symmetry | Matrices

Results from FactBites:

Rotation (mathematics) - Wikipedia, the free encyclopedia (550 words)
	In ordinary three dimensional space, a coordinate rotation can be defined by three Euler angles, or by a single angle of rotation and the direction of a vector about which to rotate.
	Rotations about the origin are most easily calculated using a 3 by 3 matrix transformation called a rotation matrix.
	Rotations about another point can be described by a 4 by 4 matrix acting on the heterogeneous coordinates.

Orthogonal matrix - Wikipedia, the free encyclopedia (2811 words)
	While it is common to describe a 3×3 rotation matrix in terms of an axis and angle, the existence of an axis is an accidental property of this dimension that applies in no other.
	A Jacobi rotation has the same form as a Givens rotation, but is used as a similarity transformation chosen to zero both off-diagonal entries of a 2×2 symmetric submatrix.
	The polar decomposition factors a matrix into a pair, one of which is the unique closest orthogonal matrix to the given matrix, or one of the closest if the given matrix is singular.

More results at FactBites »

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