(60 points) Let A=[[1,2],[2,5]].
(a) Compute the lower triangular matrix H_(1) in the Cholesky factorization A=H_(1)H_(1)^(T). Then compute the matrix A_(1)=H_(1)^(T)H_(1).
(b) Compute the lower triangular matrix H_(2) in the Cholesky factorization A_(1)= H_(2)H_(2)^(T), where A_(1) is determined in part (a). Then compute the matrix A_(2)=H_(2)^(T)H_(2).
(c) Compute Q=H_(1)H_(2)^(-T). Verify that Q is a real orthogonal matrix.
(d) Compute R=H_(2)^(T)H_(1)^(T). Verify A=QR and A_(2)=RQ.
(e) [bonus (10 extra points)] In general, suppose A is a real symmetric positive definite matrix. Let H_(1),H_(2) be the lower triangular matrices computed in the following way: (i) compute the Cholesky factorization A=H_(1)H_(1)^(T) and compute the matrix A_(1)=H_(1)^(T)H_(1), (ii) compute the Cholesky factorization A_(1)=H_(2)H_(2)^(T) and compute the matrix A_(2)=H_(2)^(T)H_(2). Set Q=H_(1)H_(2)^(-T) and R=H_(2)^(T)H_(1)^(T). Prove that Q is real orthogonal (Q^(T)Q)=(I) and R is upper triangular, and A=QR,A_(2)=RQ.
4. (60 points) Let A :
(a) Compute the lower triangular matrix H in the Cholesky factorization A = HHT Then compute the matrix A = HF H
(b) Compute the lower triangular matrix H2 in the Cholesky factorization A HH, where A, is determined in part (a). Then compute the matrix A2 = H H2
(c) Compute Q = H HT. Verify that Q is a real orthogonal matrix. (d) Compute R = HHT.Verify A = QR and A2 = RQ.
e)bonus (10 extra points] In general, suppose A is a real symmetric positive definite matrix. Let H,H be the lower triangular matrices computed in the following way: (i) compute the Cholesky factorization A = HHf and compute the matrix A= HH,ii) compute the Cholesky factorization A = HH and compute the matrix A2=HH2. Set Q = HHT and R = HHf. Prove that Q is real orthogonal (QQ = I) and R is upper triangular, and A = QR, A2 = RQ
