Primal-dual identity for ridge regression
= Primal-dual identity for ridge regression
{title2=$(X^TX+\lambda I)^{-1}X^T=X^T(XX^T+\lambda I)^{-1}$}
For any real <design matrix> $X$ and $\lambda>0$, $(X^TX+\lambda I)^{-1}X^T=X^T(XX^T+\lambda I)^{-1}$. Multiplication by $X$ identifies the primal <ridge regression> fitted values with $K(K+\lambda I)^{-1}Y$, $K=XX^T$, the <kernel-ridge hat matrix> for the <linear kernel>. No rank assumption is needed.