Let \(\mathbf{P}\) be a projection matrix. Show that the diagonal elements of \(\mathbf{P}\) all lie in the interval \([0,1]\). In particular, for \(\mathbf{P}=\mathbf{X X}^{+}\)in Theorem 5.1, the leverage value \(p_{i}:=\mathbf{P}_{i i}\) satisfies \(0 \leqslant p_{i}\) \(\leqslant 1\) for all \(i\).