18.11 Suppose that C is an n * n symmetric idempotent matrix with rank r and let V  N(0n, In).

a. Show that C = AA, where A is n * r with AA = Ir. (Hint: C is positive semidefinite and can be written as QQ, as explained in Appendix 18.1.)

b. Show that AV  N10r, Ir2.

c. Show that VCV  x2r .