Editing Infinitesimal rotation matrix (section)

== Exponential map ==
{{main|Rotation group SO(3)#Exponential map|Matrix exponential}}
Connecting the Lie algebra to the Lie group is the [[exponential map (Lie theory)|exponential map]], which is defined using the standard [[matrix exponential]]  series for {{math|''e<sup>A</sup>''}}<ref>{{Harv|Wedderburn|1934|loc=§8.02}}</ref> For any [[skew-symmetric matrix]] {{mvar|A}}, {{math|exp(''A'')}} is always a rotation matrix.{{efn|Note that this exponential  map of skew-symmetric matrices to rotation matrices is quite different from the Cayley transform discussed earlier, differing to 3rd order, 
<math>e^{2A} - \frac{I+A}{I-A} = - \frac{2}{3} A^3 +\mathrm{O}  (A^4)  ~.  </math> <br />
Conversely, a [[skew-symmetric matrix]] {{mvar|A}} specifying a rotation matrix through the Cayley map specifies the ''same'' rotation matrix through the map {{math|exp(2 artanh ''A'')}}.}}

An important practical example is the {{math|3 × 3}} case. In [[rotation group SO(3)]], it is shown that one can identify every {{math|''A'' ∈ '''so'''(3)}} with an Euler  vector {{math|1='''ω''' = ''θ'' '''u'''}}, where {{math|1='''u''' = (''x'',''y'',''z'')}} is a unit magnitude vector.

By the properties of the identification {{math|'''su'''(2) ≅ '''R'''<sup>3</sup>}}, {{math|'''u'''}} is in the null space of {{mvar|A}}. Thus, {{math|'''u'''}} is left invariant by {{math|exp(''A'')}} and is hence a rotation axis.

Using [[Rodrigues' rotation formula#Matrix notation|Rodrigues' rotation formula on matrix form]] with {{math|1=''θ'' = {{frac|''θ''|2}} + {{frac|''θ''|2}}}}, together with standard [[List of trigonometric identities#Multiple-angle and half-angle formulae|double angle formulae]] one obtains,
: <math>\begin{align}
 \exp( A ) &{}= \exp(\theta(\boldsymbol{u\cdot L}))
           = \exp \left( \left[\begin{smallmatrix} 0 & -z \theta & y \theta \\ z \theta & 0&-x \theta \\ -y \theta & x \theta & 0 \end{smallmatrix}\right] \right)= \boldsymbol{I} + 2\cos\frac{\theta}{2}\sin\frac{\theta}{2}~\boldsymbol{u\cdot L} + 2\sin^2\frac{\theta}{2} ~(\boldsymbol{u\cdot L} )^2 ,
\end{align}</math>
This is the matrix for a rotation around axis {{math|'''u'''}}  by the angle {{mvar|θ}} in half-angle form. For full detail, see [[Rotation group SO(3)#Exponential map|exponential map SO(3)]].

Notice that for infinitesimal angles second-order terms can be ignored and remains {{math|1=exp(''A'') = ''I'' + ''A''}}