[math] \require{textmacros} \def \bbeta {\bf \beta} [/math]

Recall the definitions of [math]\hat{\bbeta}_{\mbox{{\tiny MLS}}}[/math], [math]\mathbf{W}_{\lambda}[/math] and [math]\hat{\bbeta}(\lambda)[/math] from Section The ridge regression estimator . Show that, unlike the linear relations between the ridge and maximum likelihood estimators if [math]\mathbf{X}^{\top} \mathbf{X}[/math] is of full rank, [math]\hat{\bbeta}(\lambda) \not= \mathbf{W}_{\lambda} \hat{\bbeta}_{\mbox{{\tiny MLS}}}[/math] high-dimensionally.