I've read many websites, blogs, pdfs on this top but struggle to put the picture together in simple math terms, that explains how some of the terms relate to each other / are computed.

Let's assume that we use Singular Value Decomposition (SVD) to solve the PCA problem with input matrix $X$, assuming $X$ has zero mean and unit variance, the SVD of $X$:

$X = USV^{T}$

Now, $S$ is the vector of eigenvalues, sorted in decreasing order.

Can someone please explain, in math terms, how are the following terms defined?

  • e.g. Explained variance ratios = $\frac{S}{sum(S)}$
  • PCA Loadings == eigenvector of $X^{T}X$ == $V$?
  • Score matrix $T$, $T == U * Diag(S) == XV$?
  • Actual PCA components = eigenvectors of $X$?
  • Eigenvectors, can you get them from the SVD results?
  • 4
    $\begingroup$ I think I found what I am looking for in this amazing answer {stats.stackexchange.com/questions/134282/…} !! $\endgroup$
    – WillZ
    May 8, 2016 at 23:01
  • $\begingroup$ If you found everything you want, then perhaps this question should be closed as a duplicate? If there is anything you wanted but didn't find, it would be better to refocus this question on those elements alone, to avoid duplication :) $\endgroup$
    – Silverfish
    May 8, 2016 at 23:05
  • $\begingroup$ thanks, i closed this, let me think about this a bit more and maybe come back with questions if needed. $\endgroup$
    – WillZ
    May 8, 2016 at 23:35
  • 1
    $\begingroup$ @Will, feel free to ask for clarifications in my Q&A post about PCA/SVD, if anything remains unclear there. $\endgroup$
    – amoeba
    May 8, 2016 at 23:39


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