Hints for problem 3.13

Consider the singular value decomposition (SVD) of the standardized input matrix: \(\boldsymbol{X} = \boldsymbol{U}\boldsymbol{D}\boldsymbol{V}^T\). Using the columns of $\boldsymbol{V}$, note that the principal components are given by $\boldsymbol{z}_m = \boldsymbol{X} \boldsymbol{v}_m$. For the second part, show that $\hat{\beta}_{ls} = \boldsymbol{V} \boldsymbol{D}^{-1} \boldsymbol{U}^T \boldsymbol{y} = \hat{\beta}_{pcr} (p)$.