Say I am given the angular velocity of a rigid 3D object in the form $(\omega_x, \omega_y, \omega_z)^T$, where $\omega_x$ describes the angular velocity around the x-axis, $\omega_y$ around the y-axis and so on. Now from that I would like to obtain the 3D rotation axis and the angular velocity around that axis.
What I understood so far is that the rotation axis can be determined as the eigenvector with eigenvalue 1 of the infinitesimal rotation matrix $A$, which is defined as:
\begin{equation} A = \begin{pmatrix} 1 & - \omega_z & \omega_y \\ \omega_z & 1 & - \omega_x \\ -\omega_y & \omega_x & 1 \end{pmatrix} \end{equation}
How can I now compute the angular velocity around that axis?
Thanks a lot in advance!