TITLE: Revisit "Matrix exponential of a skew symmetric matrix" and rotation matrices
QUESTION [5 upvotes]: Recall the following:
Matrix exponential of a skew symmetric matrix
The conclusion is:  

$$e^C = I + \dfrac{\sin x}{x}C + \dfrac{1-\cos x}{x^2}C^2$$  


$x = \sqrt{a_1^2+a_2^2+a_3^2}$
$C=\left( \begin{array}{ccc} 
0 & -a_3 & a_2 \\
a_3 & 0 & -a_1 \\
-a_2 & a_1 & 0 \\
\end{array} \right).$    


And by the fact that $$e^C\in SO(3),$$ i.e., the exponential of skew-symmetric matrix is an element of $SO(3)$.
(Please see the reference (p.4):  http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=5766283)    

Also
The differential equation for the rotation matrix:  $$\dot{C} = -\omega^{\times}C$$ 
where $\omega^{\times} = \left( \begin{array}{ccc} 
0 & -\omega_3 & \omega_2 \\
\omega_3 & 0 & -\omega_1 \\
-\omega_2 & \omega_1 & 0 \\
\end{array} \right)$
And we can think that the rotation axis with angular velocity in this dynamic is  
$\hat{\omega} = \left( \begin{array}{ccc} 
\omega_1 \\ \omega_2 \\ \omega_3\end{array} \right)$

My question is suppose (rotation about the $z$-axis)
$\hat{\omega} = \left( \begin{array}{ccc} 
0 \\ 0 \\ \omega_3\end{array} \right)$    i.e., 
$\omega^{\times} = \left( \begin{array}{ccc} 
0 & -\omega_3 & 0 \\
\omega_3 & 0 & 0 \\
0 & 0 & 0 \\
\end{array} \right)$  
How to derive the corresponding rotation matrix is:  
$R=\left( \begin{array}{ccc} 
\cos\omega t  & -\sin\omega t  & 0 \\
\sin\omega t   & \cos\omega t & 0 \\
0 & 0 & 1 \\
\end{array} \right).$  
There is an identity matrix $I$ in the equation; how to deal with that?

REPLY [3 votes]: For rotation about the $e_3$, or z-axis, let 
$$\eqalign{
 B &= \omega^\times\,t \cr
 b &= \omega_3\,t \cr
 G_3 &= B/b \cr
}$$
$G_3$ is a skew-symmetric (0-1) matrix, whose square is
$$\eqalign{
 G_3^2 &= e_3e_3^T-I \cr\cr
}$$
Evaluating the exponential formula
$$\eqalign{
 R = e^B &= \frac{\sin b}{b}\,B &+\,I + \frac{1-\cos b}{b^2}\,B^2 \cr
     &= (\sin b)\,G &+\,I + (1-\cos b)\,G^2 \cr
     &= (\sin b)\,G &+\,(\cos b)\,I + (1-\cos b)\,e_3e_3^T \cr
\cr}$$
The first 2 terms yield the familiar $(2\times 2)$ cosine-sine rotation matrix. 
The third term sets the lower right element by subtracting the $\cos(b)$ factor coming from the $(3\times 3)$ identity matrix in the second term, and replacing it with a value of 1.