[SOLVED] scala ECS130 Homework Assignment 5 Due: 4:00pm, February 27, 2019

ECS130 Homework Assignment 5 Due: 4:00pm, February 27, 2019
1. Let A be nonsingular, and let 1 2 n > 0 be its singular values.
(a) Find the SVD of A1 in terms of the SVD of A. What are the singular values and singular vectors of A1?
(b) Deduce that A1 = 1 and cond(A) = A A1 = / , 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 (1, u1, v1)?
5. (a) Let A Rnn be symmetric and positive definite. Determine the SVD of A. (b) Let A Rnn be symmetric and indefinite. Determine the SVD of A.
6. Let A Rnn 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 = rei 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
1