Page 430 - Matrix Analysis & Applied Linear Algebra

426 Chapter 5 Norms, Inner Products, and Orthogonality

5.12.6. Prove that if σ r is the smallest nonzero singular value of A m×n , then

σ r = min Ax =1/ A †
,

2
x 2 =1 2
x∈R(A T )
which is the generalization of (5.12.5).

5.12.7. Generalized Condition Number. Extend the bound in (5.12.8) to
include singular and rectangular matrices by showing that if x and
˜ x are the respective minimum 2-norm solutions of consistent systems
˜
Ax = b and A˜ x = b = b − e, then

κ −1 e ≤ x − ˜ x ≤ κ e , where κ = A A †
.

b x b
Can the same reasoning given in Example 5.12.1 be used to argue that
for 2 , the upper and lower bounds are attainable for every A?

2
5.12.8. Prove that if | | <σ for the smallest nonzero singular value of A m×n ,
r
T
T
T
then (A A + I) −1 exists, and lim →0 (A A + I) −1 A = A .
†
5.12.9. Consider a system Ax = b in which

.835 .667
A = ,
.333 .266
and suppose b is subject to an uncertainty e. Using ∞-norms, deter-
mine the directions of b and e that give rise to the worst-case scenario
in (5.12.8) in the sense that x − ˜ x / x = κ ∞ e / b .
∞ ∞ ∞ ∞

5.12.10. An ill-conditioned matrix is suspected when a small pivot u ii emerges

during the LU factorization of A because U −1 =1/u ii is then
ii
large, and this opens the possibility of A −1 = U −1 L −1 having large
entries. Unfortunately, this is not an absolute test, and no guarantees
about conditioning can be made from the pivots alone.
(a) Construct an example of a matrix that is well conditioned but
has a small pivot.
(b) Construct an example of a matrix that is ill conditioned but has
no small pivots.