Unpenalized intercept in ridge regression ($\hat{\alpha }=\overline{Y}-{\overline{x}}^T\hat{\beta }$)

In ridge regression, penalize slopes but ordinarily leave the regression intercept free. Centering gives $\hat{\beta }=(X_c^T{X}_c+\lambda I{)}^{-1}{X}_c^T{Y}_c$ and $\hat{\alpha }=\overline{Y}-{\overline{x}}^T\hat{\beta }$. An exactly centered predictor matrix has $\hat{\alpha }=\overline{Y}$ for every penalty. A table with a penalty-dependent intercept therefore cannot arise from exactly centered columns under this convention.