Suppose there are two vector sets $A$ and $B \in \mathbb{R}^{f \times 3}$
If there is an orthogonal matrix $R \in \mathbb{R}^{3 \times 3}$ can align there two vector set perfectly, e.g. $RA = B$
What is the necessary condition for A and B to satisfy that?
I know the orthogonal matrix can be solved by the orthogonal Procrustes algorithm, but that only gives the solution. I want to know the condition for A and B to make them be perfectly aligned by the $R$.
Thanks.
Since, if $R$ is an orthogonal matrix and $v\in\mathbb R^3$, $\lVert Rv\rVert=\lVert v\rVert$, a necessary condition is that $\lVert A\rVert=\lVert B\rVert$. And it turns out that this condition is sufficient too. That is, two vectors $A,B\in\mathbb R^3$ have the same norm if and only if there is some orthonormal matrix $R$ such that $RA=B$.