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I am computing the principle component matrix of a financial database and to obtain the second factor I extrapolate the second vector. So far, it's easy, but I wonder, do the signs of the second component have to be different? I mean can there be only positive (or negative) values ? Regards

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  • $\begingroup$ Btw., it's "principal", not "principle". $\endgroup$ – A. Donda Mar 26 '15 at 15:34
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    $\begingroup$ Perhaps stats.stackexchange.com/questions/38571 answers this question: it demonstrates that provided all the correlations are positive, then indeed all the coefficients of the first PC have the same sign and some coefficients of the second PC must have different signs. But what exactly do you mean by "signs ... be different"? Obviously the signs cannot match one-for-one, for then the inner product of the two PCs could not be zero, yet it must be zero. $\endgroup$ – whuber Mar 26 '15 at 15:37
  • $\begingroup$ C. Chris, did my answer help you? $\endgroup$ – A. Donda May 30 '15 at 16:17
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Every subsequent eigenvector from a PCA is constrained to be orthogonal to all previous eigenvectors. If your variables tend to be mainly positively correlated, this will lead to an all-positive first eigenvector. In order to be orthogonal to that the second eigenvector will have to have a mixture of signs.

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