An infinitesimal rotation matrix or differential rotation matrix is a matrix representing an infinitely small rotation.
While a rotation matrix is an orthogonal matrix representing an element of (the special orthogonal group), the differential of a rotation is a skew-symmetric matrix in the tangent space (the special orthogonal Lie algebra), which is not itself a rotation matrix.
An infinitesimal rotation matrix has the form
where is the identity matrix, is vanishingly small, and
For example, if representing an infinitesimal three-dimensional rotation about the x-axis, a basis element of
The computation rules for infinitesimal rotation matrices are as usual except that infinitesimals of second order are routinely dropped. With these rules, these matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals.[1] It turns out that the order in which infinitesimal rotations are applied is irrelevant.
^(Goldstein, Poole & Safko 2002, §4.8)
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An infinitesimalrotationmatrix or differential rotationmatrix is a matrix representing an infinitely small rotation. While a rotationmatrix is an orthogonal...
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