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I thought the singular value decomposition was a generalized form of the spectral theorem for non-square matrices. That being said, what exactly are the sizes of these matrices here? Because if A is m x n, then P must be n x n for the spectral decomposition, which leaves Q to be m x n. Then, the spectral decomposition will force V to be n x n, so U = QV comes out to be m x n which is not necessarily a rotation.


How can I get the QP out of A? I am confused and my brain stops thinking about matrixes.


I only see rotation and scaling here. Is it possible to recover a shear transformation in the decomposition? And does this decomposition imply that a shear is just a combination of a rotation and a scaling transformation? (This may be the answer to a question I had on an earlier slide...)


Does the rotation matrix V have to be orthonormal?


Can we get a more detailed, slower explanation on this slide during class? So many matrices my brain just left T_T. Are V and D the eigenvectors and eigenvalues?


I'm having a hard time visualizing/understanding how U = QV eliminates possible reflections in the case where only Q is applied. Can someone explain this?


For every matrix, will the polar decomposition be unique?


I get how we can decompose a transformation into its components, but I still can’t seem to think of a good application as to why we would ever want to decompose it.


What is polar decomposition?


What does rotation/reflection mean here? The rotation relative to what?


Is there any method that we can have choose better values in U that could expedite the computation


Are polar decomposition and spectral decomposition frequently used in graphics, I feel like most of the time we are building the matrix A instead of decomposing it into components.


How exactly are the matrices being applied to the 3D objects? Or maybe how is the cow being represented by matrices?