For some application it turned out that a convex sum of particular rotation matrices in 3D had a real eigenvector. Each rotation matrix had a (different) time dependent rotation angle, but the eigenvector of the sum was not time dependent (at least it did not seem so, this had to be done numerically). It is a very complicated sum so I will not reproduce it here. But it made me start to experiment with sums of rotation matrices, and I have the following question:
For two rotation matrices $R(u_1,\theta_1)$ and $R(u_2,\theta_2)$ in $SO(3)$, with rotation axis $u_i$ and rotation angle $\theta_i$ I want to determine what are the eigenvectors of $R_1+R_2$, in terms of $u_i$ and $\theta_i$. I tried to put this in mathematica but got huge expressions that do not simplify down. This does not seem like something too crazy to be interested in, has it been done before? I tried to write it as $$ (R_1+R_2)\bar{v} = \lambda \bar{v} $$ then set $\bar{v}=\bar{u}_i+\bar{w_i}$ where $\bar{u}_i$ is the real eigenvector of $R_i$ and $w_i$ is orthogonal to it. It does not help much though, I just then have to solve $$ \bar{u}_1+\bar{u}_2 + R_1\bar{w}_1 + R_2\bar{w}_2 = \lambda \bar{v} \ . $$ I wonder if there is a "nice" expression for the eigenvector, preferably in terms of rotation axis and angles of the matrices? This will help, I suppose, to determine when the eigenvector is independent of the angle of the rotation matrices. I know that when the two rotation axis are the same the eigenvector is the rotation axis. Is it the ONLY case?