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Concept
Infinitesimal rotation matrix
Summary
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. It turns out that the order in which infinitesimal rotations are applied is irrelevant.
An infinitesimal rotation matrix is a skew-symmetric matrix where:
As any rotation matrix has a single real eigenvalue, which is equal to +1, the corresponding eigenvector defines the rotation axis.
Its module defines an infinitesimal angular displacement.
The shape of the matrix is as follows:
We can introduce here the associated infinitesimal rotation tensor:
Such that its associated rotation matrix is . When it is divided by the time, this will yield the angular velocity vector.
These matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals . To understand what this means, consider
First, test the orthogonality condition, QTQ = I. The product is
differing from an identity matrix by second order infinitesimals, discarded here. So, to first order, an infinitesimal rotation matrix is an orthogonal matrix.
Next, examine the square of the matrix,
Again discarding second order effects, note that the angle simply doubles.
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