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I asked this question before on MSE but go no answers. It seems that the problem is rather difficult so I thought of trying here. Given two matrices $A,B\in SO(n)$, each describing a rotation by angles $\alpha$ and $\beta$ about axis $\bar{a},\bar{b}$, when does the sum $A+B$ have a real eigenvector independent of $\alpha,\beta$?

I know that if $[A,B]=0$ then their sum has has a real eigenvector independent of their rotation angles; this is because they share eigenspaces, their rotation axis is the same, and the sum $A+B$ has an angle independent eigenvector equal to the (joint) rotation axis. But is this the only case?

Also, what if the matrices just general SO(n) elements?

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    $\begingroup$ If $n>3$, it makes no sense to say that matrices describe rotations about axes: it only makes sense to say that they describe rotations inside planes (2-dimensional subspaces). Also, you should be aware that this is no longer the general form of $SO(n)$ matrices: they can simultaneously rotate inside two or more orthogonal planes, with independent angles for each plane. See e.g. this wikipedia article. $\endgroup$
    – Stefano Gogioso
    Commented Sep 5, 2018 at 11:10
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    $\begingroup$ For an example of conditions under which such a real eigenvector exists independently of the angles, let $n\geq 5$ and let $A,B$ be rotations on a single plane $P_A,P_B$ each (by which I mean that they are the identity on the $(n-2)$-dimensional subspaces $P_A^\bot,P_B^\bot$). Then such eigenvectors always exist: any vector in the subspace $W:= P_A^\bot \cap P_B^\bot$, which is at least $(n-4)$-dimensional, is a real eigenvector of $A+B$ with eigenvalue 2. To see this, choose diagonalisations of both $A$ and $B$ which agree on $W$. $\endgroup$
    – Stefano Gogioso
    Commented Sep 5, 2018 at 11:11
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    $\begingroup$ You can make this argument arbitrarily general, by the way: if $A$ rotates about $k_A$ orthogonal planes and $B$ rotates about $k_B$ orthogonal planes, then taking $n \geq 2k_A+2k_B+1$ always guarantees the existence of at least on eigenvector for $A+B$ independent of all rotation angles. $\endgroup$
    – Stefano Gogioso
    Commented Sep 5, 2018 at 11:14
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    $\begingroup$ You are right, I was sloppy asking about the SO(n) case. This, in the SO(3) case is then the requirement that the matrices commute. However, is this the only possibility? I.e. A+B has a real eigenvector if and only if $W$ is non-empty? $\endgroup$
    – myorbs
    Commented Sep 5, 2018 at 11:31
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    $\begingroup$ Duplicate: math.stackexchange.com/questions/2883868/… $\endgroup$
    – Steven Landsburg
    Commented Sep 5, 2018 at 14:12

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