Infinitesimal rotation matrix

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

While a rotation matrix is an orthogonal matrix ${\displaystyle R^{\mathsf {T}}=R^{-1}}$ representing an element of ${\displaystyle SO(n)}$ (the special orthogonal group), the differential of a rotation is a skew-symmetric matrix ${\displaystyle A^{\mathsf {T}}=-A}$ in the tangent space ${\displaystyle {\mathfrak {so}}(n)}$ (the special orthogonal Lie algebra), which is not itself a rotation matrix.

An infinitesimal rotation matrix has the form

${\displaystyle I+d\theta \,A,}$

where ${\displaystyle I}$ is the identity matrix, ${\displaystyle d\theta }$ is vanishingly small, and ${\displaystyle A\in {\mathfrak {so}}(n).}$

For example, if ${\displaystyle A=L_{x},}$ representing an infinitesimal three-dimensional rotation about the x-axis, a basis element of ${\displaystyle {\mathfrak {so}}(3),}$

${\displaystyle dL_{x}={\begin{bmatrix}1&0&0\\0&1&-d\theta \\0&d\theta &1\end{bmatrix}}.}$

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.