Let $\vec x, \vec y \in \mathbb{R}^3$ and $\bf A $ be a $3 \times 3$ real matrix. Under what conditions does $\bf A$ distribute over a cross product:
$$ \mathbf{A} (\vec x \times \vec y) = (\mathbf{A}\vec x) \times (\mathbf{A} \vec y) $$
The cross product in this case is the vanilla one over $\mathbb{R}^3$: $$ (\vec a \times \vec b)_i = \epsilon_{ijk} a_j b_k$$
Where $\epsilon$ is the Levi-Civita symbol. My suspicion is matrices where $\mathbf{A}^{-1} = \mathbf{A}^T$ like the rotations in $\mathbb{R}^3$ would satisfy this condition. I can imagine the equality holding when $\vec x$ is aligned with the axis of rotation (i.e. $\mathbf{A}\vec x = \vec x$) and therefore rotating $\vec y$ also rotates $\vec x \times \vec y$.
Would anyone back the general case up with a proof for me?
