I was wondering if I could get a mathematical description of the relationship between the singular value decomposition (SVD) and the principal component analysis (PCA).

To be more specific I have some point which I don't understand very well, at least from a mathematical point of view.

What are the principal components (PCs), bearing in mind that we are using the SVD to compute the PCA?

Which part of the SVD becomes the PCs?

What is the relationship between the orthogonal matrices from the SVD plus the diagonal matrix with the

*scores*and*loadings*of the PCA?I have read that the principal components can describe big data sets with very few

*loadings*vectors, how is this (mathematically) possible? In which way can these few principal components tell me something about the*variance*of a big data set and what does the SVD has to do with this process?

I have tried to be very specific making my questions, if something is not so clear I apologize.

Thanks in advance for your help!

PS I have made my homework and look very close to : What is the intuitive relationship between SVD and PCA? but I could not fine the answers I am looking for. I believe mine are more related to the mathematical concepts than the practical ones. Anyways, if you believe this questions is unnecessary (duplicate) I will remove it.