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 .
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(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.
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