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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?

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  • $\begingroup$ Of course immediately after posting I found this question here: math.stackexchange.com/questions/279173/… Should I delete this question or allow it to me marked duplicate? $\endgroup$ – cms Dec 13 '18 at 22:58
  • $\begingroup$ Yes, absolutely, DELETE! $\endgroup$ – Cosmas Zachos Jan 4 at 2:22

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