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I am computing the PCA projection matrix of some data. I notice that the elements of first basis vector (corresponding to the highest eigenvalue) are always positive. My data is real and contain both positive and negative values. Can someone please explain the reason and/or point out related literature?

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    $\begingroup$ The basis elements are defined only up to sign. It sounds like the algorithm is choosing the sign that makes the initial component positive. See stats.stackexchange.com/search?q=pca+sign+eigenvector for related threads. Now, if it's the case that all elements of the first component are positive, that is usually because all your variables are strongly positively correlated. $\endgroup$ – whuber Feb 27 at 12:41
  • $\begingroup$ @whuber: Yes. It's true that my data points are highly correlated. Could you please give pointer to some related papers? $\endgroup$ – talk2speech Feb 27 at 12:46
  • $\begingroup$ I doubt there are any papers devoted to this phenomenon. $\endgroup$ – whuber Feb 27 at 15:59
  • $\begingroup$ @whuber: Thanks for the pointer to the correlation. This phenomenon can be explained by Perron–Frobenius theorem en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem $\endgroup$ – talk2speech Mar 4 at 14:59
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    $\begingroup$ Yes, that's a good find: it supplies rigor to support the intuition and experience. $\endgroup$ – whuber Mar 4 at 15:16

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