Closed-form ridge regression estimator
= Closed-form ridge regression estimator
{title2=$\widehat\beta_\lambda=(X^TX+\lambda I)^{-1}X^TY$}
For $\lambda>0$, the <ridge regression> objective $\lVert Y-X\beta\rVert_2^2+\lambda\lVert\beta\rVert_2^2$ has <gradient> $2(X^TX+\lambda I)\beta-2X^TY$. The matrix $X^TX+\lambda I$ is <positive-definite matrix>[positive definite], so the unique minimizer is $\widehat\beta_\lambda=(X^TX+\lambda I)^{-1}X^TY$.