Rotation matrix and orthogonal matrix

Question

If I am given an orthogonal matrix $A$ and I right multiply by $W $ to get $B=AW$ where $W$ is a diagonal matrix then essentially, I am scaling the columns of $A$ by the diagonals of $W$, preserving orthogonality. My question is, am I able to construct a rotation matrix $R $ that transforms $A$ to $B$? If so, how? My intuition is SVD but I can't quite see it.

Answer 1

[This was a comment that got a bit too long] The answer to your question will depend on your definition of a "rotation" in $\mathbb R^n$. In $\mathbb R^3$ a rotation is an orientation preserving element of $\mathrm{O}_3$. It has eigenvalues $\{1,\exp(i\theta),\exp(-i\theta)\}$, hence if $B=AW$, and you want, I think, a rotation matrix $R$ with $B=RAR^{-1}= RAR^{\intercal}$. But then $B$ is an orthogonal matrix, and so since $W=A^{-1}B$, $W$ must be orthogonal also, and hence (as $W=W^\intercal$) we must have $W^2=I$, and $W \in \{\mathrm{diag}(\pm 1, \pm 1, \pm 1)\}$.

Thus it appears the situation you wish to consider is a very special one.