Problem 2.2 Assume p S n and let X E Rm'? be a fullrank matrixl, and H = X(XTX)'1XT (1) Note that H is a. square 77. X '11 matrix. This problem is devoted to understanding the properties H. (a) Show that H is symmetric (HT = H) and idempotent (H2 = H). Hint: (AB)? : BTAT. Use this to derive (ABCV : OTBTAT. Also (A'1)T : (A7)\Any matrix that satises conditions in (a) is an orthogonal projection matrix. In this problem, we will verify this directly for the H given in (1). Let V = Im(X). (b) Show that for any y E RP, Hy E V. (e) Let e = y 7 Hy = (I 7 H)y and show that e E Vi. Hint: Use [Im(X)]J' : ker(XT) The combination of parts (b) and (c) show that Hy is the projection of y onto V. (d) Show that I 7 H is symmetric and idempotent (hence an orthogonal projection matrix). Can you guess what subspace it projects onto? Since H is symmetric, it has a spectral decomposition of the form H = U AUT, Where U is orthogonal and A is diagonal. (e) Show that A2 : A. What does this tell you about the eigenvalues of H ? Pick any vector y 6 R\" and let ft] : Hy be the projection of 1/ onto Im(X) and e : (I 7 H )y the error (or residual). (U Show that Iii/H2 = Illl2 + IIGHZ- Hmt: Recall that Hyuz : yTy, carpand and use properties ofH and I 7 H. (g) Show that for any y 6 IR\