Set
A = rand(8, 4) * rand(4, 6), [U, D, V]= svd(A)
(a) What is the rank of A? Use the column vectors of V to generate two matrices V l and V2 whose columns form orthonormal bases for R(AT) and N(A), respectively. Set
p = V2 * V2ʹ, r = P * rand(6, 1), w = Aʹ * rand(8, 1)
If r and w had been computed in exact arithmetic, they would be orthogonal. Why? Explain. Use MATLAB to compute rTw.
(b) Use the column vectors of U to generate two matrices U1 and U2 whose column vectors form orthonormal bases for R(A) and N(AT), respectively. Set
Q = U2*U2ʹ, y = Q* rand(8. 1), z = A * rand(6, 1)
Explain why y and z would be orthogonal if all computations were done in exact arithmetic. Use MATLAB to compute yTz.
(c) Set X = pinv(A). Use MATLAB to verify the four Penrose conditions:
(i) AXA = A
(ii) XAX = X
(iii) (AX)T = AX
(iv) (XA)T = XA