Can one use PCA scores to check for multivariate normality of the data? How can you check multivariate normality, using the scores from the PCA (principal component analysis)? Or what can we expect about the scores if the data is multivariate normal distributed?
 A: If the original data is multivariate normal then the PCA scores will be as well (any linear combination of normals is normal).  But the reverse is not guaranteed.
In fact I would expect that something that is not multivariate normal (but not hugely different) would still result in PCA scores (at least some of them) that looked more normal than the original due to the same ideas as the Central Limit Theorem (there may be a specific CLT for this).  Since the PCA scores are in a sense a weighted average of other variables I would expect them to be more normal than the original.
There exist examples of multivariate distributions that have Normal margins, but are not Multivariate Normal.  The PCA scores are a rotation of this data, but the rotation will still likely have normal margins (so the individual PCA scores would look normal) but still be non multivariate normal in the same way (just rotated).
So I would not be comfortable saying anything about the multivariate normality of a dataset (other than possibly "normality is not ruled out") based on the PCA scores, no matter how normal they may look.
