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.

bobzhangyc

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

gloose

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...)

jonasjiang

Does the rotation matrix V have to be orthonormal?

corgo

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?

tianez

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?

Starboy

For every matrix, will the polar decomposition be unique?

TejasFX

I get how we can decompose a transformation into its components, but I still canâ€™t seem to think of a good application as to why we would ever want to decompose it.

shough

What is polar decomposition?

BlueCat

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

Concurrensee

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

Oh_skr

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.

shoes

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

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 canâ€™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?