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Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
A quaternion that is slightly off still represents a rotation after being normalized: a matrix that is slightly off may not be orthogonal any more and is harder to convert back to a proper orthogonal matrix. Quaternions also avoid a phenomenon called gimbal lock which can result when, for example in pitch/yaw/roll rotational systems, the pitch ...
The quaternion elements vary continuously over the unit sphere in ℝ 4, (denoted by S 3) as the orientation changes, avoiding discontinuous jumps (inherent to three-dimensional parameterizations) Expression of the rotation matrix in terms of quaternion parameters involves no trigonometric functions
There are at least two ways of representing quaternions as matrices in such a way that quaternion addition and multiplication correspond to matrix addition and matrix multiplication. One is to use 2 × 2 complex matrices, and the other is to use 4 × 4 real matrices.
It turns out that g ∈ SO(3) represented in this way by Π u (g) can be expressed as a matrix Π u (g) ∈ SU(2) (where the notation is recycled to use the same name for the matrix as for the transformation of it represents). To identify this matrix, consider first a rotation g φ about the z-axis through an angle φ,
is the rotation matrix through an angle θ counterclockwise about the axis k, and I the 3 × 3 identity matrix. [4] This matrix R is an element of the rotation group SO(3) of ℝ 3 , and K is an element of the Lie algebra s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} generating that Lie group (note that K is skew-symmetric, which characterizes ...
which is a quaternion of unit length (or versor) since ‖ ‖ = + + + = Most importantly, the above equations for composition of rotations are precisely the equations for multiplication of quaternions =. In other words, the group of unit quaternions with multiplication, modulo the negative sign, is isomorphic to the group of rotations with ...
Given a unit vector in 3 dimensions, for example (a, b, c), one takes a dot product with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector. The eigenvectors of that spin matrix are the spinors for spin-1/2 oriented in the direction given by the vector. Example: u = (0.8, -0.6, 0) is a unit vector ...