TITLE: Find the rotation axis and angle of a matrix
QUESTION [15 upvotes]: $$A=\frac{1}{9}
\begin{pmatrix}
-7 & 4 & 4\\ 
4 & -1 & 8\\ 
4 & 8 & -1
\end{pmatrix}$$
How do I prove that A is a rotation ? How do I find the rotation axis and the rotation angle ?

REPLY [10 votes]: The simplest way to find the rotation angle is to take the trace of the matrix, the sum of the diagonal elements. By Cameron Buie's answer this equals $1 + 2\cos(\theta)$ where $\theta$ is the angle of rotation. $\theta$ can then be determined up to sign which will depend on the orientation of the axis of rotation chosen.
For non-symmetric matrices, the axis of rotation can be obtained from the skew-symmetric part of the rotation matrix, $S = .5(R-R^\mathrm{T})$;
Then if $S=(a_{ij})$, the rotation axis with magnitude $\sin\theta$ is $(a_{21},a_{02},a_{10})$.