In Kernel PCA, the kernel trick works because we can show that there is an equivalency between eigenvectors of the kernel matrix and eigenvectors of the covariance matrix. I know the math to go from one to the other but there is still one point that puzzles me very much:
These matrices have potentially very different dimensions, so I don't completely understand how the equivalence works. Let's say the covariance matrix is $(d\times d)$ dimensional and the kernel matrix $(N \times N)$, that means we have $N$ eigenvectors for the kernel matrix. Applying the formula to compute the eigenvectors of the covariance matrix from the kernel's eigenvectors would give $N$ vectors in total. Are some of the computed vectors identical? I don't understand how to reconcile the dimensions here.