Interpreting positive and negative signs of the elements of PCA eigenvectors If I center my variables and then run a PCA, do I need to interpret negative eigenvectors different than positive eigenvectors?
Clarification: In my PCA I have in a component both negative and positive variables. This is probably very basic, but I've been told different things about the interpretation of the variables within a component, so I just need some clarification. Is it so that the negative values within a component moves in the same direction and the positive in the opposite? Or is it so that I should only look at the absolute values of the variables within a component?
 A: I think you have it backwards.  If the value is positive, then a higher score on that variable is associated with a higher score on the component, if the value is negative, then a higher score implies a lower score on the component.  
In addition, people sometimes use PCA to determine whether to keep or combine certain variables for a subsequent analysis.  This is not, strictly speaking, an appropriate use of PCA.  Factor analysis should be used for this purpose, but at any rate, people do it.  In such a case, people will look at the absolute value to see if it is above some arbitrary threshold, such as .5, and if so, retain (or combine), and if not, drop.  For what it's worth, I don't recommend this.  
Update: I can't tell if I answered the right question or not.  @whuber's second comment, in my opinion, is right on the money, and also consistent with my first paragraph above.  However, the question is now different than before, and different from how I understand @whuber's comment, so I am a little confused.  Essentially, PCA solves for the eigenvectors and eigenvalues.  Neither will be negative whether or not you centered your variables first.  The eigenvalues are the lengths of the corresponding eigenvectors.  Just as I cannot buy a board -10 feet (i.e., -3 meters) long to build a patio, you cannot have a negative eigenvalue.  The eigenvector returned will also be positive.  You could negate it by multiplying all the signs by -1, but as @whuber notes, that would be meaningless.  Once again as @whuber notes, the relative signs are meaningful, and their relation to the component is as I stated in my first paragraph above.  That is, the relative signs (negative vs. positive) will denote the same relationship between higher (/ lower) scores on the variable and the component whether the variables were centered first or not.
A: centering your variables shouldn't change the PCA results, as PCA first determines a correlation matrix and goes on from there. The correlations between your variables should be the same regardless, so the PCA results should not be affected by any mean centering you perform.
A: When we say correlation that means can be two directional i.e. positive and negative. Interpretation of the principal components is based on finding which variables are most strongly correlated with each component, i.e., which of these numbers are large in magnitude, the farthest from zero in either positive or negative direction.
