Background: I'm having trouble using principal component analysis to try and align two data sets. I have two sets of 3D point data, and I can use PCA to get principal axes of the two sets of data. I do this by finding the eigenvectors of the covariance matrix for each set of data. This gives me two sets of principal axes defined by 6 eigenvectors. PCA gives me an eigenvalue that says my data corresponds strongly along an axis parallel to the eigenvector and through the centroid of the data. Therefore, the eigenvector could be pointing in the opposite direction and the axis would still be the same. I want to know the significance of the direction in which the eigenvector is pointing (what is the difference between the eigenvector, and that eigenvector * -1). Problem: I'm trying to align the two data sets using PCA -- but I can't do this if the corresponding direction vectors are allowed to point in the opposite direction. So if the one of the axes 'x' from data set 1 is pointing close to a higher elevation, and axes 'x' from data set 2 is pointing in the opposite direction, I'm in trouble... I hope this makes some sense.