ECS130 Homework Assignment 5 Due: 4:00pm, February 27, 2019
1. Let A be nonsingular, and let σ1 ≥ σ2 ≥ ··· ≥ σn > 0 be its singular values.
(a) Find the SVD of A−1 in terms of the SVD of A. What are the singular values and singular vectors of A−1?
(b) Deduce that ∥A−1∥ = σ−1 and cond(A) = ∥A∥ ∥A−1∥ = σ /σ , where cond(A) is the 2n 221n
condition number of A.
2. Ex 7.1
3. Ex 7.3
4. If A = abT , where a ∈ Rm and b ∈ Rn, what is the first (largest) singular triplet (σ1, u1, v1)?
5. (a) Let A ∈ Rn×n be symmetric and positive definite. Determine the SVD of A. (b) Let A ∈ Rn×n be symmetric and indefinite. Determine the SVD of A.
6. Let A ∈ Rn×n of full rank. Use the SVD to determine a polar decomposition of A, i.e., A=QP whereQisorthogonal,andP =PT >0.
Note: (1) this is analogous to the polar form z = reiθ of a complex scalar z, where i = √−1. (2) Inspired to learn more about the polar decomposition. Try the problems in Exercise 7.8. (3) The plar decomposition has wide applications, such as animation.
7. Ex. 7.10 (“Latent semantic analysis”) — option
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