 Because we are never working directly in the feature space, the kernel-formulation of PCA is restricted in that it computes not the principal components themselves, but the projections of our data onto those components.To evaluate the projection from a point in the feature space formula_17 onto the kth principal component formula_19 (where superscript k means the component k, not powers of k)

We note that formula_21 denotes dot product, which is simply the elements of the kernel formula_22.It seems all that's left is to calculate and normalize the formula_23, which can be done by solving the eigenvector equation

where N is the number of data points in the set, and formula_25 and formula_26 are the eigenvalues and eigenvectors of K. Then to normalize the eigenvectors formula_27's, we require that

Care must be taken regarding the fact that, whether or not formula_29 has zero-mean in its original space, it is not guaranteed to be centered in the feature space (which we never compute explicitly).