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Kernel ridge regression minimizes empirical squared error plus the squared norm in a Reproducing-kernel Hilbert space. The representer theorem reduces it to ridge regression with the kernel matrix as its feature Gram matrix.

Because the columns of the design matrix are centred, ridge regression with an unpenalized intercept solvesThe normal equations give, for $\lambda >0$,The fitted values are consequentlyIf the objective is normalized by $n$, the same formulas hold after replacing $\lambda$ by $n\lambda$.

The ridge regression estimator minimizesIts gradient vanishes exactly when $(X^{T}X+\lambda I)\beta =X^{T}Y$. Since $\lambda >0$ makes this matrix positive definite,

The ridge regression estimator is the elastic net at $\alpha =0$, while the lasso regression estimator is the elastic net at $\alpha =1$.