 Of course, this formula_12 creates linearly independent vectors, so there is no covariance on which to perform eigendecomposition "explicitly" as we would in linear PCA.Instead, in kernel PCA, a non-trivial, arbitrary formula_12 function is 'chosen' that is never calculated explicitly, allowing the possibility to use very-high-dimensional formula_12's if we never have to actually evaluate the data in that space.Since we generally try to avoid working in the formula_12-space, which we will call the 'feature space', we can create the N-by-N kernel

which represents the inner product space (see Gramian matrix) of the otherwise intractable feature space.