344 CHAPTER 14. NUMERICAL SOLUTIONS OF LINEAR SYSTEMS
7. If you are considering a system of the form Ax = b and A−1 does not exist, will eitherthe Gauss Seidel or Jacobi methods work? Explain. What does this indicate aboutusing either of these methods for finding eigenvectors for a given eigenvalue?
8. Verify that∥x∥
∞≡max{|xi| , i = 1, · · · ,n : x = (x1, · · · ,xn)}
is a norm. Next verify that
∥x∥1 ≡n
∑i=1|xi| , x = (x1, · · · ,xn)
is also a norm on Fn.
9. Let A be an n× n matrix. Denote by ∥A∥2 the operator norm taken with respect tothe usual norm on Fn. Show that
∥A∥2 = σ1
where σ1 is the largest singular value. Next explain why∥∥A−1
∥∥2 = 1/σn where σn is
the smallest singular value of A. Explain why the condition number reduces to σ1/σn
if the operator norm is defined in terms of the usual norm, |x|=(
∑nj=1
∣∣x j∣∣2)1/2
.
10. Let p,q > 1 and 1/p+1/q = 1. Consider the following picture.
b
a
x
t
x = t p−1
t = xq−1
Using elementary calculus, verify that for a,b > 0,
ap
p+
bq
q≥ ab.
11. ↑For p > 1, the p norm on Fn is defined by
∥x∥p ≡
(n
∑k=1|xk|p
)1/p
In fact, this is a norm and this will be shown in this and the next problem. Using theabove problem in the context stated there where p,q > 1 and 1/p+1/q = 1, verifyHolder’s inequality
n
∑k=1|xk| |yk| ≤ ∥x∥p ∥y∥q
Hint: You ought to consider the following.
n
∑k=1
|xk|∥x∥p
|yk|∥y∥q
Now use the result of the above problem.