rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix...

infinitesimal rotation matrix or differential rotation matrix is a matrix representing an infinitely small rotation. While a rotation matrix is an orthogonal...

The product of two rotation matrices is a rotation matrix, and the product of two reflection matrices is also a rotation matrix. Regardless of the dimension...

of the rotation matrix associated with an eigenvalue of 1. As long as the rotation angle α {\displaystyle \alpha } is nonzero (i.e., the rotation is not...

the rotation matrix are not all independent—as Euler's rotation theorem dictates, the rotation matrix has only three degrees of freedom. The rotation matrix...

every rotation can be represented uniquely by an orthogonal matrix with unit determinant. Moreover, since composition of rotations corresponds to matrix multiplication...

with the rotation matrix method. There are three basic approaches to rotating a vector v→: Compute the matrix product of a 3 × 3 rotation matrix R and the...

over the complex numbers. For example, this is the case for a generic rotation matrix. Many results for diagonalizable matrices hold only over an algebraically...

SO(3) Rotations and reflections in two dimensions CORDIC Infinitesimal rotation matrix Irrational rotation Orientation (geometry) Rodrigues' rotation formula...

rotations, starting from a known standard orientation. Equivalently, any rotation matrix R can be decomposed as a product of three elemental rotation...

corner, which must be set to 1. For example, the counter-clockwise rotation matrix from above becomes: [ cos ⁡ θ − sin ⁡ θ 0 sin ⁡ θ cos ⁡ θ 0 0 0 1 ]...

such as shearing or rotation. For example, if R {\displaystyle R} is a square matrix representing a rotation (rotation matrix) and v {\displaystyle...

Kabsch and Shinji Umeyama, is a method for calculating the optimal rotation matrix that minimizes the RMSD (root mean squared deviation) between two paired...

are related by a rotation about some fixed axis. This also means that the product of two rotation matrices is again a rotation matrix and that for a non-identity...

In computer vision a camera matrix or (camera) projection matrix is a 3 × 4 {\displaystyle 3\times 4} matrix which describes the mapping of a pinhole camera...

while he was working at Argonne National Laboratory. A Givens rotation acting on a matrix from the left is a row operation, moving data between rows but...

of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3), the group of all rotation matrices...

inertia matrix is a constant real symmetric matrix. A real symmetric matrix has the eigendecomposition into the product of a rotation matrix Q {\displaystyle...

consisting of the 3x3 rotation matrix R and the 1x3 translation vector p. The matrix is augmented to create a 4x4 square matrix. g s t ( 0 ) = [ R p 0...

element matrix in an effort to simplify analysis. The transformation is equivalent to the product of the Clarke transformation and a rotation. The Park...

{\displaystyle 3\times 3} rotation matrix and t {\displaystyle \mathbf {t} } is a 3-dimensional translation vector. The essential matrix is then defined as:...

generate any orthonormal matrix of the space, whose determinant is +1. Due to the non-commutativity of the matrix product, the rotation system must be ordered...

composition of a boost and a rotation. This rotation is called Thomas rotation, Thomas–Wigner rotation or Wigner rotation. If a sequence of non-collinear...

mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization...

representing the unit rotation axis, and an angle, θ ∈ R, an equivalent rotation matrix R is given as follows, where K is the cross product matrix of ω, that is...

complex matrix into a rotation, followed by a rescaling followed by another rotation. It generalizes the eigendecomposition of a square normal matrix with...

\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}},} which is the standard matrix equation of a rotation of axes in two dimensions. The inverse transformation is or...

representing complex numbers e i θ {\displaystyle e^{i\theta }} as the rotation matrix, that is, e i θ = ( cos ⁡ θ − sin ⁡ θ sin ⁡ θ cos ⁡ θ ) = cos ⁡ θ (...

}}=(\omega _{x},\omega _{y},\omega _{z})} . This is an infinitesimal rotation matrix. The linear mapping Ω acts as a cross product ( ω × ) {\displaystyle...

Mathematically, if everything in the universe undergoes a rotation described by a rotation matrix R, so that a displacement vector x is transformed to x′...