Editing 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 {{mvar|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.<ref group="nb">Note that this exponential map of skew-symmetric matrices to rotation matrices is quite different from the Cayley transform discussed earlier, differing to the third order,
:<math>e^{2A} - \frac{I+A}{I-A}=- \tfrac{2}{3} A^3 +\mathrm{O} \left(A^4\right) . </math>
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'')}}.</ref>

An important practical example is the {{nowrap|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>\mathbf{su}(2) \cong \mathbb{R}^3</math>, {{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.

According to [[Rodrigues' rotation formula#Matrix notation|Rodrigues' rotation formula on matrix form]], one obtains,

:<math>\begin{align}
 \exp( A ) &= \exp\bigl(\theta(\mathbf{u}\cdot\mathbf{L})\bigr) \\
           &= \exp \left( \begin{bmatrix} 0 & -z \theta & y \theta \\ z \theta & 0&-x \theta \\ -y \theta & x \theta & 0 \end{bmatrix} \right) \\
           &= I + \sin \theta \ \mathbf{u}\cdot\mathbf{L} + (1-\cos \theta)(\mathbf{u}\cdot\mathbf{L} )^2 ,
\end{align}</math>

where
: <math> \mathbf{u}\cdot\mathbf{L} = \begin{bmatrix} 0 & -z & y \\ z & 0&-x \\ -y & x & 0 \end{bmatrix} .</math>

This is the matrix for a rotation around axis {{math|'''u'''}} by the angle {{mvar|θ}}. For full detail, see [[Rotation group SO(3)#Exponential map|exponential map SO(3)]].