Math 726  University of South Carolina
D. Meade  Spring 1996

Assignment #2: Norms and the SVD

Due: Friday, 9 February 1996

\underline{Problems from Golub and VanLoan}:

Section Page Problems
2.5  74 # 2.5.2

# 2.5.8

2.7  84 # 2.7.1

(Compute ∥I∥ in all matrix norms we have discussed.)

Other Problems

1. Graph the sets {x ∈ R^{2} : ∥x∥ = 1} for the 1-, 2-, 4-, and ∞-norms. (These are the unit balls for the different norms.)

2. Graph the set {x ∈ R^{2} : ∑_{i=1}^{2} x_{i}^{1/2} = 1}. Why is there no 1/2-norm? (Which of the defining properties of a norm is/are not satisfied?)

3. The version of showsvd that I wrote adds a rank-one update to the image (matrix) at each step. Generalize the showsvd function so that at each step the image is updated by a matrix with rank specified by the user. Also, modify the titles to indicate relative error (in the 2-norm) between the approximate and original images.

4. Use your new version of showsvd to determine the lowest rank approximation which reproduces the original image with a 10%, 1%, and 0.1% relative error (in the 2-norm). How much information is needed to produce each of these approximations?

\underline{Notes}

• Try to be as efficient as possible in your implementation of showsvd. In particular, use block matrix operations whenever possible. Matlab commands for counting, and timing, operations

include flops and cputime.

• If you have any questions, please ask in class, in office hours, by e-mail, . . ..