The vector spaces N(A), R(A), N(AT), R(AT) are the four fundamental subspaces associated with a matrix A. We can use MATLAB to construct orthonormal bases for each of the fundamental subspaces associated with a given matrix. We can then construct projection matrices corresponding to each subspace.
(a) Set
A = rand(5, 2) * r and(2, 5)
What would you expect the rank and nullity of A to be? Explain. Use MATLAB to check your answer by computing rank(A) and Z = null(A). The columns of Z form an orthonormal basis for N(A).
(b) Next set
Q = orth(A), W = null(A'), S = [Q W]
The matrix 5 should be orthogonal. Why? Explain. Compute S * S' and compare your result to eye(5). In theory, ATW and WTA should both consist entirely of zeros. Why? Explain. Use MATLAB to compute ATW and WTA.
(c) Prove that if Q and W had been computed in exact arithmetic then we would have
I-WWT = QQT and QQTA = A
[Write SST in terms of Q and W.] Use MATLAB to verify these identities.
(d) Prove that if Q had been calculated in exact arithmetic then we would have QQTb = b for all b ∈ R(A). Use MATLAB to verify this by setting b = A * rand(5, 1) and then computing Q * Q' * b and comparing it to b.
(e) Since the column vectors of Q form an orthonormal basis for R(A), it follows that QQT is the projection matrix corresponding to R(A). Thus, for any c ∈ R, the vector q = QQTc is the projection of c onto R(A). Set c = rand(5, 1) and compute the projection vector q. The vector r = c - q should be in N(AT). Why? Explain. Use MATLAB to compute A' * r.
(f) The matrix WWT is the projection matrix corresponding to N(AT). Use MATLAB to compute the projection w = WWTc of c onto N(AT) and compare the result to r.
(g) Set Y = orth(A') and use it to compute the projection matrix U corresponding to R(AT). Let b = rand(5. 1) and compute the projection vector y = U * b of b onto R(AT). Compute also U * y and compare it to y. The vector s = b - y should be in N(A). Why? Explain. Use MATLAB to compute A * s.
(h) Use the matrix Z = null(A) to compute the projection matrix V corresponding to N(A). Compute V * b and compare it to s.