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I applied PCA on an $n\times p$ matrix where $n$ means the number of samples and $p$ means the number of variables, and I am using the 1st principal component (PC) as the new representation of the entire dataset. I confirmed that the 1st PC has the highest variance among all PCs, as expected. But I don't know how to interpret the values in the 1st PC, while before applying PCA a negative or positive value had a meaning. I observed that row means of the original data has a $-0.99$ correlation with the 1st PC, meaning that if a sample had a positive value across all variables, then this sample had a highly negative value in the 1st PC, while a consistent negative value across variables in the original data corresponds to a highly positive sample in the 1st PC. This is extremely confusing. How can I interpret that behavior?

I also realized that all loadings for the 1st PC are negative. I think this might be related to what I observe. Any thoughts?

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marked as duplicate by Nick Cox, Stephan Kolassa, mdewey, jbowman, Michael Chernick Jan 23 '18 at 15:54

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  • $\begingroup$ So, you have in view an alternative reduction of the data, the (equally weighted) average of the original variables. That could be simpler to think about and to explain to others and just as helpful downstream. $\endgroup$ – Nick Cox Jan 23 '18 at 10:05
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PCA gives principal vectors pointing in the best directions to project your data onto, in terms of variance or squared error. It ignores the sign. A principal vector pointing in the opposite direction is also a valid solution to PCA but will give you principal components with the opposite sign. If you want, you can manually invert the sign for a specific principal component for all your data.

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