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Just a hypothetical question: What happens if I do a second principal component analysis over the principal components derived from the first principal component analysis?

What will be the difference between the eigenvectors derived from the first and second PCA? Will there be any loss of information?

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    $\begingroup$ Since your first PCA identifies orthogonal vectors, your second PCA should in principle do nothing (since it should basically find the same axes as the first round). I don't know enough to say whether that principle holds in practice i.e. if there are practical/algorithmic reasons why you might find different behaviour. $\endgroup$ – mkt Jul 18 '17 at 12:56
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    $\begingroup$ But expect arbitrariness as to sign, i.e. a PC might have reversed coefficient signs in the second solution. $\endgroup$ – Nick Cox Jul 18 '17 at 13:50
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Turning my comment above into an answer:

Since your first PCA identifies orthogonal vectors, your second PCA should in principle do nothing (since it should basically find the same axes as the first round). But as @NickCox points out, coefficients might be reversed.

However, there may be small differences in practice for algorithmic reasons.

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