 Closed-form ridge regression estimator

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 ${\hat{\beta }}_{\lambda }=(X^{T}X+\lambda I{)}^{-1}{X}^{T}Y$.