4. Consider the hat matrix H, and let hij be the (ij)th entry. (a) Show that hij=xiT(XTX)1xjT=xjT(XTX)1xiT=hji, where xi is the ith row of X. (b) Let X=UVT be the SVD of X. Show that hij=uiTuj. (c) Show that i=1nhij=j=1nhij=1. (d) Show that n1hiir1, where r is the number of rows in X that are exactly the same as xi 4. Consider the hat matrix H, and let hij be the (ij)th entry. (a) Show that hij=xiT(XTX)1xjT=xjT(XTX)1xiT=hji, where xi is the ith row of X. (b) Let X=UVT be the SVD of X. Show that hij=uiTuj. (c) Show that i=1nhij=j=1nhij=1. (d) Show that n1hiir1, where r is the number of rows in X that are exactly the same as xi