# Difficulty understanding the dimension differences in kernel PCA

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.