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I know similar questions have been asked before already but I'm still a bit confused about the understanding of 'loadings'.

Eigenvectors are unit-scaled loadings; and they are the coefficients (the cosines) of orthogonal transformation (rotation) of variables into principal components or back. (Loadings vs eigenvectors in PCA: when to use one or another?)

[...]cosϕ is the Pearson correlation between V and component F1 (What is the proper association measure of a variable with a PCA component (on a biplot / loading plot)?)

So the eigenvectors indicate the correlation between the component and the variables and loadings are those eigenvectors multiplied by the square root of the corresponding eigenvalue. The eigenvalue equals the variance explained by the component.

Now lets say we perform a correlation-matrix based PCA.

If we PCA-analyzed not just centered but standardized (centered then unit-variance scaled) variables the three variables vectors (not their projections on the plane) would be of the same, unit length. Then it follows that a loading is correlation, not covariance, between a variable and a component.[...] In correlation-based PCA a1=cosϕ because h=1, but principal components are not those same principal components as we get from covariances-based PCA. (see prior link)

For me this sounds like in this case the loading equals the eigenvector. But when the loading is computed as mentioned above then it has to be different from the eigenvector (unless sqrt(eigenvalue) = 1). It appears that on several websites eigenvectors and loadings are used interchangeably and loadings are often defined as "the correlation between a variable and a component" but I'm not sure anymore if the eigenvector or loading (as computed) is meant.

What exactly does a loading value tell me regarding a variable? How does that differ from what the eigenvector value tells me about that variable?
Also, can a loading for a variable on correlation based PCA be greater than 1?

(I'm sorry if I'm mixing things up here but the more I read into it the more I get confused so I hope you can help me getting a clearer understanding. Help is much appreciated!)

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    $\begingroup$ I don't have time right now to answer everything here (maybe later), but "So the eigenvectors indicate the correlation between the component and the variables" - this is wrong. Where do you see this in the quotes that you gave? Correlations are given by the elements of loadings, not of eigenvectors. But you are right that the terminology is field-dependent and some people say "loadings" to refer to [unit-length] "eigenvectors". $\endgroup$ – amoeba Oct 17 '16 at 16:38
  • $\begingroup$ I thought that the cosϕ between a variable and a component is equal to the eigenvector value for that variable and thus equals the correlation between the variable and the component. So the eigenvector values just tell me the direction of the corresponding component? Thanks for clearing that misunderstanding of mine already! $\endgroup$ – mri Oct 17 '16 at 20:14
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    $\begingroup$ I thought that the cosϕ between a variable and a component is equal to the eigenvector I've added a paragraph about eigenvector element to my answer, the 2nd one you cite. The tricky word is "cosine" which may mean both "cosine of rotation" and "cosine similarity measure", dependent on the context. $\endgroup$ – ttnphns Oct 17 '16 at 21:20
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    $\begingroup$ Just to echo amoeba's remark: "So the eigenvectors indicate the correlation between the component and the variables" - this is wrong. Eigenvector values are not association measures, they are coefficients of transformation. $\endgroup$ – ttnphns Oct 17 '16 at 21:54

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