Editing Infinitesimal rotation matrix (section)

An '''infinitesimal rotation matrix''' or '''differential rotation matrix''' is a [[matrix (mathematics)|matrix]] representing an [[infinitesimal|infinitely]] small [[rotation]].

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

An infinitesimal rotation matrix has the form

:<math> I + d\theta \, A,</math>

where <math>I</math> is the identity matrix, <math>d\theta</math> is vanishingly small, and <math>A \in \mathfrak{so}(n).</math>

For example, if <math>A = L_x,</math> representing an infinitesimal three-dimensional rotation about the {{mvar|x}}-axis, a basis element of <math>\mathfrak{so}(3),</math> then

:<math> L_{x} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{bmatrix} </math>,

and 

:<math> I+d\theta L_{x} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -d\theta \\ 0 & d\theta & 1 \end{bmatrix}. </math>

The computation rules for infinitesimal rotation matrices are the usual ones except that infinitesimals of second order are dropped. With these rules, these matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals.<ref>{{Harv|Goldstein|Poole|Safko|2002|loc=§4.8}}</ref> It turns out that ''the order in which infinitesimal rotations are applied is irrelevant''.