For kernel PCA, I see that people usually center feature maps - hence center the kernel matrix. But for kernel ridge regression, it seems that we do not need to center the feature maps/ kernel matrix for the regression.
It is intuitive for me that the linear regression will $\hat{y}(x) = w^{T}\phi(x)$ will give the same results with or without centering $\phi(x)$ (by adjusting the parameters $w$). But I'm trying to understand it from the kernel point of view: the corresponding kernel ridge regression is of the form $\hat{y}(x) = k_x^{T}(K+\lambda I)^{-1}y$.
Assuming without ridge regularization for simplicity, is it correct to say that $\hat{y}(x) = k_x^{T}K^{-1}y$ is the same as $\hat{y}(x) = \tilde{k}_x^{T} \tilde{K} ^{-1}y$, where $\tilde{k}_x$ and $\tilde{K}$ are the centered version of $k_x$ and $K$? Is there a way to mathematically prove it?