# Kernel PCA and K largest eigenvectors

How can one prove that the optimal kPCA solution $$a^*=\{a_1...a_K\}$$ are the $$k$$-largest Eigenvectors of the (centered) kernel matrix $$K$$?

I referred to a lot of resources and couldn't find a proper explanation.

• We aim to choose a direction in which all of the features have the highest variance. why is that? because features low variance doesn't really have that much influence in predicting the outcome, because it does not vary that much for different samples. Hence, we choose the direction with the largest amount of variance for all the features that's also more informative. In order to choose the direction with the highest variance, we choose the eigenvectors as directions that lead to highest variance. – Fatemeh Asgarinejad Jul 15 '19 at 1:04

An indirect way would be the ratio of variance in all the dimensions if we know the eigenvalues corresponding to eigenvectors. So if an eigenvector has eigenvalues $$e1$$ then the ratio contributed by it will be $$e1\over (e1+e2+e3+e4.... +en)$$. The largest eigenvector will have the largest eigenvalue.