PCA fits a high-dimensional ellipse to a cloud of points. The axes of the ellipse correspond to the eigenvectors. The length of each axes correspond to the eige

PCA fits a high-dimensional ellipse to a cloud of points. The axes of the ellipse correspond to the eigenvectors. The length of each axes correspond to the eigenvalues. You can reduce dimensions by saying some of the eigenvalues are "noise" and discarding them.