Visualizing SVD/PCA and applying to new data

I have an SVD for a data set as U S V'. My data are images, but my questions will be general.

I know how to reduce the rank of the original data by zeroing out small singular values and computing U*S*V', but:

1. I want to isolate one dimension of variance, say, the one corresponding to the n th largest eigenvalue. This should be some basis vector, right? How do I get that basis vector? I thought it would be the n th column of U*S, but it's not.
2. How do I fit some new data that wasn't in the original set to the SVD's bases?

Thanks in advance.

Answer 1

[Chaman Sabharawal]

Assuming your data matrix A is observation vs arrtibutes. You are reducing attrubutes. Both ways reduction standard A= USV^T or AV=US Reducing it on attributes only you get A reduced to AV. For Reducing on one dimension replace V with a desired direction vector. I hope this is what you are looking for. Chaman