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 righthanded coordinate system into a lefthanded 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. ...
Properties
Let be a general rotation matrix of any dimension:  The dot product of two vectors remains unchanged when both are operated upon by a rotation matrix:


 where 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.
 Any rotation matrix can be represented as the exponential of a skewsymmetric matrix A:

 where the exponential is defined in terms of its Taylor series and 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 skewsymmetric 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 skewsymmetric (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 counterclockwise rotation. The matrix to rotate a column vector in cartesian coordinates about the origin by a counterclockwise 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. ...
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 nonzero 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 skewsymmetric 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 righthanded cartesian x, y and zaxes 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 xaxis is defined as:
 where θ_{x} is the roll angle.
 Rotation around the yaxis is defined as:
 where θ_{y} is the pitch angle.
 Rotation around the zaxis is defined as:
 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 3dimensional rotation matrix 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.  is a rotation matrix in
The set of all rotations in , 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 3dimensional 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. ...
AngleAxis representation and quaternion representation 
In three dimensions, a rotation can be defined by a single angle of rotation, θ, and the direction of a unit vector, , 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. ...
This rotation may be simply expressed in terms of its generator: 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. ...
The angleaxis 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. ...
where i, j, and k are the three imaginary parts of Q.
AngleAxis 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 representing a rotation about unit axis for angle θ is given by: where is a unit tensor of second order, which is a sum of three dyadics , where are three orthogonal unit vectors of any orthonormal reference frame. The given representation does not depend on the actual current orientation of reference frame because the unit tensor itself has the same representation in any orthonormal reference frame (nonorthonormal reference frames will be considered just few lines later).
The Rodrigues' rotation formula simply follows from the above representation as soon as 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 is responsible for a component of the vector , which is parallel to the axis of rotation and is not affected by the multiplication . The length of this component is , where r is the length of the vector and α is the angle between vectors and .
The projector gives us a component of the vector , which is exactly orthogonal to . The length of this component is . This component is then scaled by cosθ depending on the actual rotation angle θ.
And the last part of the expression for the rotation tensor is responsible for a component of the final vector , which is orthogonal to both and as soon as . The length of vector is also equal due to definition of the crossproduct of two vectors. For the crossed product in algebra and functional analysis, see crossed product. ...
As a result the three parts , and of the rotation tensot construct a local orthogonal reference frame which is most convinient for description of the actual rotation of any given vector .
The above representation can be is generalized onto the case of nonorthonormal reference frame by constructing the unit tensor as (assuming Einstein summation), where are covectors of vectors . 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 are build out of as: where each triplet {i,j,k} is a cyclic permutations of {1,2,3} triplet. In orthonormal reference frames the vectors coincide with their "co"counterparts .
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 "zxz" Euler angles, in righthanded 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. ...
carrying out the multiplications yields: 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 where the columns of span the space orthogonal to q and G is the Givens rotation of θ degrees, i.e. See also In linear algebra and geometry, a coordinate rotation is a type of transformation from one system of coordinates to another system of coordinates such that distance between any two points remains invariant under the transformation. ...
In geometry a rotation representation expresses the orientation of an object (or coordinate frame) relative to a coordinate reference frame. ...
In mathematics, an isometry, isometric isomorphism or congruence mapping is a distancepreserving isomorphism between metric spaces. ...
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 geometry, Rodrigues rotation formula is a vector formula for a rotation in space, given its axis. ...
A sphere rotating around its axis. ...
In mechanics and geometry, the rotation group is the set of all rotations about the origin of 3dimensional Euclidean space, R3. ...
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. ...
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