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369 HW9 1. Let A be an n × n matrix with eigenvalues λ1 , . . . , λn . Let A = QR be the QR-factorization of A and write rii for the (i, i)th entry of R. Show that λ1 . . . λn = ±r11 . . . rnn . 2. Let 1 1 0 A = −2 1 0 0 0 1 Let A = QR be the QR-factorization of A. Without doing the Gram-Schmidt procedure, state the (1, 3) and (2, 3) entries of R. Explain your reasoning. 3. Let A = (v1 |v2 ), where 1 v1 = 1 , 0 1 v2 = 0 . 1 (a) Use Gram-Schmidt to find orthonormal vectors u1 , u2 with the same span as v1 , v2 . (b) Find a unit vector u3 which is orthogonal to both u1 and u2 . (c) Let Q = (u1 |u2 |u3 ) compute R = Qt A. (This is QR factorization for non-square matrix.) 4. Let A, u1 , u2 be the same as in Problem 3 and define Q = (u1 |u2 ), R = Qt A. Let 2 4 . b= −3 (a) Verify that Ax = b has no solutions. (b) Solve Rx = Qt b and check that Ax ≈ b. 1