Develop the following results associated with the Hat matrix: Show that H and H, are symmetric and idempotent with ranks + 1 and m. respectively.

b. Use the partitioned form of the inverse to develop the relation betwem the centered and uncentered Hat matrices given in (6.7).

c. Show that if the design matrix contains J, then HJ-J. Also show that Hr=0

d. Let X denote the mean vector, computed without the ith case, and let X denote the centered design matrix without the ith row. Define the squared distance from case i to the rest of the data as d=(-Z) (X)(-a).

and show that = (17)(1) Him: Show that -(-2), and apply (6.16) to the centered design matrix.