Math 191 Wilkening
Spring 2023
Homework 7
due Mon, Mar. 20, 11:59 PM
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1. (3 points) (Variant of I.8.5, page 68, Strang.) Show that AT has the same (nonzero) singular values as A and conclude that ∥A∥ = ∥AT ∥. Provide a counterexample to demonstrate that ∥Ax∥ need not equal ∥AT x∥ for all vectors x. Then show that ∥Ax∥ = ∥AT x∥ for all vectors x if ATA = AAT.
2. (2 points) (Variant of I.8.8, page 68, Strang.) Compute the SVD
A=0 0 3=USVT,
i.e., find U, V and S. (For this matrix, U and V are permutation matrices).
3. (2 points) (I.8.12, page 69, Strang.) Find the SVD of the rank 1 matrix 2 4
Factor AT A into QΛQT .
4. (1 points) (I.8.17, page 70, Strang.) If v is an eigenvector of AT A with λ ̸= 0, then ——— is an eigenvector of AAT . (Justify your answer.)
5. (2 points) (I.8.21, page 70, Strang.) Show that if A has rank r with nonzero singular values σ1 ≥···≥σr >0thenthesingularvaluesofAATAareσj3,1≤j≤r.
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6. (3 points) (II.2.2, page 135, Strang.) Why do A and A+ have the same rank? If A is square, do A and A+ have the same eigenvectors? What are the eigenvalues of A+?
7. (3 points) (Variant of II.2.4, page 135, Strang.) Which matrices have A+ = A? (Look at A+A.)
8. (4 points) (II.2.5, page 135, Strang.) Suppose A has independent columns (rank r = n; nullspace = zero vector)
(a) Describe the m × n matrix S in A = USV T . How many nonzero entries in S? (b) Show that ST S is invertible by finding its inverse.
(c) Write down the n × m matrix (ST S)−1ST and identify it as S+.
(d) Substitute A = U S V T into (AT A)−1 AT and identify that matrix as A+ .
The conclusion is that if A has rank n, then ATAxˆ = ATb leads to A+ = (ATA)−1AT.
9. (3 points) (II.2.11, page 135, Strang.) Show that if Q is m×n and QTQ = I then QT = Q+. If A = QR for invertible R, show that QQT = AA+.
10. (3 points) (I.9.2, page 80, Strang.) Find a closest rank-1 approximation to each of the following matrices in the 2-norm:
3 0 3 2 1 A1= 2 , A2= 2 0 , A3= 1 2 .
11. (2 points) (I.9.3, page 80, Strang.) Find a closest rank-1 approximation in the L2 norm to cosθ −sinθ
12. (2 points) (I.9.10, page 80, Strang.) If A is a 2 × 2 matrix with σ1 ≥ σ2 > 0, find ∥A−1∥2 and ∥A−1∥2F .
A= sinθ cosθ .
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