As Kernel PCA is the same as PCA in higher dimension space, shouldn't the eigenvectors obtained be orthogonal?
Suppose, I have $n$ data points and let $a$ and $b$ be two eigenvectors of covariance matrix of mapped data, and $\alpha \in \mathbb{R}^n$ and $\beta \in \mathbb{R}^n$ be the corresponding eigenvectors of Kernel PCA problem (obtained by doing eigenanalysis of kernel matrix $K$). Then $a$ and $b$ can be written as linear combination of input data, i.e. $$a = \sum_{i=1}^n \alpha_i \phi(x_i)$$ and $$b = \sum_{j=1}^n \beta_j \phi(x_j).$$
Their inner product is: $$\langle a,b \rangle \,=\, \left\langle \sum_{i=1}^n \alpha_i \phi(x_i) ,\sum_{j=1}^n \beta_j \phi(x_j)\right\rangle \, = \, \sum_{i=1}^n\sum_{j=1}^n \alpha_i \beta_j K_{ij}.$$
Why is this value zero?