Question: Let A be an m x n matrix with rank A = n. (a) Prove that the matrix P = A(ATA)~l AT is a projection

Let A be an m x n matrix with rank A = n.
(a) Prove that the matrix P = A(ATA)~l AT is a projection matrix, meaning that P2 = P, cf. Exercise 2.5.8.
(b) Construct the projection matrix corresponding to
(i)

(ii)

(iii)

(iv)

(c) Prove that P is symmetric.
(d) Prove that mg P = mg A.
(e) Show that v* = P b is the closest point on the subspace mg A - rng P to b.
(f) Show that if A is nonsingular, then P = I. How do you interpret this in light of part (e)?

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