Question

V. QUESTION 5 full Exercise 5.3.29 of [Moon and Stirling, 2000]: If A e Cmkn where m > n and A has column rank, the QR factorization of A is Ri Here, Q, e Cmn, Q2 E Cmx(m-), and R e Cnm. Also, the columns of Q, are orthogonal. Show the following: 1) "Skinny" factorization: A-QRi

Answer 1

Given that A elementof c^m times n: m > n e(A) = n A = [Q_1, Q_2][R_1 0]: Q_1 elementof c^m times n Q_2 elementof c^m times_(m - n) R_1 elementof c^m times n A = [Q_1 R_1 + Q_2 times 0] = Q_1 R_1 A = Q_1 R_1 A = Q_1 R_1 c^m times n = [c^m times n] [c^n times n] c^m times n = [c^m times n][: {A^m times n] times [B^n times P] = [AB]^m times n] L.H.S = R.H.S Hence e(A) = e(Q_1 R_1) = e(Q_1) A^T = [[R_1 Q_2][R,0]]^T = [R,0][Q_1 Q_2] A^T = R_1 Q_1 + 0 times Q_2 = R_1 Q_1 c^n times m = [c^n times m] [c^m times n] = c^n times n Hence e^(A^T) = n/(e A^T) = e(Q_2): e(R_1 Q_1) = n