Axis and Angle of Rotation of 3x3 matrix \begin{pmatrix}
√ 3/2 & -1/4 & √3/4\\ 
1/2 & √3/4 & -3/4\\ 
0 & √3/2 & 1/2
\end{pmatrix}
How do I find the axis and angle of rotation of this matrix?
 A: The trace of a rotational matrix equals $1+2\cos\phi$, so this extracts the angle. If you take the antisymmetric part of the matrix:
$$A-A^T$$
it looks like
$$\begin{bmatrix}0&-z&y\\z&0&-x \\-y &x &0\end{bmatrix}$$
where $(x,y,z)$ is the axis (not normalized). This vector is actually the axis, multiplied by $2\sin\phi$, which defines the sense of rotation (the sign of the angle).
The inverse of this transformation is the Rodrigues' formula:
http://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula
Of course your matrix first has to be a pure rotational matrix! If it's not, you have to decide what you want anyway. For instance, if you want to present your transformation as a composition of rotation and scaling, you take the polar decomposition.
A: *

*Find the axis $x$ solving the equation $$
Ax=x
$$

*Find the angle looking at $Ax^\perp_1,Ax^\perp_2$ where $x^\perp_{1,2}$ both are orthogonal to $x$ and $x^\perp_1\perp x^\perp_2$


NB: note that you can get $\cos\theta = \frac 12(tr A -1)$ directly, but you don't get the sign of $\theta$.
