Closed-form ridge regression estimator (${\hat{\beta }}_{\lambda }=(X^{T}X+\lambda I{)}^{-1}{X}^{T}Y$)

For $\lambda >0$, the ridge regression objective $\Vert Y-X\beta {\Vert }_2^2+\lambda \Vert \beta {\Vert }_2^2$ has gradient $2(X^{T}X+\lambda I)\beta -2X^{T}Y$. The matrix $X^{T}X+\lambda I$ is positive definite, so the unique minimizer is ${\hat{\beta }}_{\lambda }=(X^{T}X+\lambda I{)}^{-1}{X}^{T}Y$.

In an eigenbasis of $G=X^{T}X$, ridge regression divides the corresponding component of $X^{T}Y$ by $\gamma +\lambda$. If $\gamma =0$, its eigenvector $v$ satisfies $Xv=0$, and hence $v^T{X}^{T}Y=0$ exactly. Only positive eigenvalues contribute to the limit, giving the Moore-Penrose inverse expression for the minimum-norm least-squares solution. No full-rank or sample-size assumption is needed.

For any real design matrix $X$ and $\lambda >0$, $(X^{T}X+\lambda I{)}^{-1}{X}^T=X^T(X{X}^T+\lambda I{)}^{-1}$. Multiplication by $X$ identifies the primal ridge regression fitted values with $K(K+\lambda I{)}^{-1}Y$, $K=X{X}^T$, the kernel-ridge hat matrix for the linear kernel. No rank assumption is needed.

For $X^{T}X=V\Lambda V^T$, ridge regression multiplies the fitted response along the $i$th normalized sample principal component by the shrinkage factor ${\Lambda }_{ii}/({\Lambda }_{ii}+\lambda )$. Directions with variance much larger than $\lambda$ are nearly retained, while directions with variance much smaller than $\lambda$ are nearly removed.