In the textbook I am studying Multiple View Geometry in Computer Vision, Second Edition. Richard Hartley, Andrew Zisserman. pg 40, it is stated that
The affine matrix $A$ can always be decomposed as $$R(\theta)R(-\phi)DR(\phi)$$ where $R(-\phi)$ and $R(\phi)$ are rotations by $\theta$ and $\phi$ respectively and $D$ is the diagonal matrix. This decomposition follows directly from the SVD: writing $A = UDV^T = (UV^T)(VDV^T) = R(\theta)(R(-\phi)DR(\phi))$ since $U$ and $V$ are orthogonal matrices.
So I have a couple questions.
- I thought orthonormal matrices such as $V$ are not just rotational matrices but also can be reflection matrices? If so, why does the decomposition above state that they are just rotation matrices?
- What is the point in decomposing it into 3 rotational matrices? Couldn't it just be decomposed as $A = UDV^T = R(\theta)DR(\phi)$?