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    $\begingroup$ Is it possible that there exist different conventions in different fields here? I stumbled over this question, because in my field (chemometrics) the usual way is to have orthonormal loadings. In other words, the scale/magnitude/$\sqrt{{\rm eigenvalues}}$ goes into the scores, not into the loadings. Loadings equal the inverse = transpose of the eigenvector matrix. I double checked this with both the "Handbook of Chemometrics and Qualimetrics" and the "Comprehensive Chemometics" which I consider the 2 most important reference works for chemometrics. $\endgroup$ Commented Mar 29, 2015 at 18:19
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    $\begingroup$ Side note: In chemometrics, calculating scores from original data is of huge importance, as lots of predictive models use PCA rotation (!) for pre-processing, so the limited use of loadings is IMHO our main use for PCA. $\endgroup$ Commented Mar 29, 2015 at 18:21
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    $\begingroup$ @cbeleites, It is not only possible that PCA/FA terminologic conventions may differ in different fields (or in different software or books) - I state they do differ. In psychology and human behaviour "loadings" are usually what I labeled by the name (loadings are very important in those fields because interpretation of the latents is pending, while the scores may be scaled down, standardized, and nobody cares). On the other hand, many R users on this site have called PCA's eigenvectors "loadings" which might probably come from the function documentation. $\endgroup$ Commented Mar 29, 2015 at 19:32
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    $\begingroup$ You must be mixing up. When you properly compute PC scores with the help of loadings you end up with simply standardized components. You do not compute these scores by the same formula as you do with eigenvectors; rather, you should use formulas described in the link of my #4. $\endgroup$ Commented Apr 5, 2015 at 13:35
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    $\begingroup$ @AshlinJP, Imagine a 2d example: you are rotating a cartesian system, axes X and Y (these axes are a pair of variables) into new (also orhogonal) axes I and II (the two pr. components). If the angle of rotation is very small, i.e. the cosine is close to 1, then the new axes almost coincide with th old ones; I, for example, almost coinsides with X, which is alias to say that "X contributes to I much while Y contributes to I small". $\endgroup$ Commented Jul 2, 2020 at 19:40