(the special orthogonal group), the differential of a rotation is a skew-symmetric matrix
(the special orthogonal Lie algebra), which is not itself a rotation matrix.
representing an infinitesimal three-dimensional rotation about the x-axis, a basis element of
[1] It turns out that the order in which infinitesimal rotations are applied is irrelevant.
Dividing it by the time difference yields the angular velocity tensor: These matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals.
[2] 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.
Next, examine the square of the matrix, Again discarding second-order effects, note that the angle simply doubles.
This hints at the most essential difference in behavior, which we can exhibit with the assistance of a second infinitesimal rotation, Compare the products dAx dAy to dAy dAx, Since
is second-order, we discard it: thus, to first order, multiplication of infinitesimal rotation matrices is commutative.
In other words, the order in which infinitesimal rotations are applied is irrelevant.
This useful fact makes, for example, derivation of rigid body rotation relatively simple.
But one must always be careful to distinguish (the first-order treatment of) these infinitesimal rotation matrices from both finite rotation matrices and from Lie algebra elements.
When contrasting the behavior of finite rotation matrices in the Baker–Campbell–Hausdorff formula above with that of infinitesimal rotation matrices, where all the commutator terms will be second-order infinitesimals, one finds a bona fide vector space.
Technically, this dismissal of any second-order terms amounts to Group contraction.
Approximating Δθ as θ/N, where N is a large number, a rotation of θ about the axis may be represented as: It can be seen that Euler's theorem essentially states that all rotations may be represented in this form.
The product Aθ is the "generator" of the particular rotation, being the vector (x, y, z) associated with the matrix A.
This shows that the rotation matrix and the axis-angle format are related by the exponential function.
One can derive a simple expression for the generator G. One starts with an arbitrary plane[3] defined by a pair of perpendicular unit vectors a and b.
This modified rotation matrix can be rewritten as an exponential function.
Analysis is often easier in terms of these generators, rather than the full rotation matrix.
Analysis in terms of the generators is known as the Lie algebra of the rotation group.
Using Rodrigues' rotation formula on matrix form with θ = θ⁄2 + θ⁄2, together with standard double angle formulae one obtains, This is the matrix for a rotation around axis u by the angle θ in half-angle form.
Notice that for infinitesimal angles second-order terms can be ignored and remains exp(A) = I + A Skew-symmetric matrices over the field of real numbers form the tangent space to the real orthogonal group
at the identity matrix; formally, the special orthogonal Lie algebra.
In this sense, then, skew-symmetric matrices can be thought of as infinitesimal rotations.
Another way of saying this is that the space of skew-symmetric matrices forms the Lie algebra
Moreover, since the exponential map of a connected compact Lie group is always surjective, it turns out that every orthogonal matrix with unit determinant can be written as the exponential of some skew-symmetric matrix.
the exponential representation for an orthogonal matrix reduces to the well-known polar form of a complex number of unit modulus.
is odd; since each single block of order 2 is also an orthogonal matrix, it admits an exponential form.