Page 134 - Matrix Analysis & Applied Linear Algebra

128 Chapter 3 Matrix Algebra

so the relative change in the solution is

% & % &
x − ˜ x $ $ $ $ B B
< $ A −1 $ B = A −1 $ A = κ . (3.8.8)
$
∼
x A A
Again, the condition number κ is pivotal because when κ is small, a small
relative change in A cannot produce a large relative change in x, but for larger
values of κ, a small relative change in A can possibly result in a large relative
change in x. Below is a summary of these observations.

Sensitivity and Conditioning

• A nonsingular matrix A is said to be ill conditioned if a small
relative change in A can cause a large relative change in A −1 .
The degree of ill-conditioning is gauged by a condition number
κ = A A −1 , where , is a matrix norm.

• The sensitivity of the solution of Ax = b to perturbations (or
errors) in A is measured by the extent to which A is an ill-
conditioned matrix. More is said in Example 5.12.1 on p. 414.

Example 3.8.2
It was demonstrated in Example 1.6.1 that the system

.835x + .667y = .168,
.333x + .266y = .067,

is sensitive to small perturbations. We can understand this in the current context
by examining the condition number of the coefficient matrix. If the matrix norm
(3.8.7) is employed with

.835 .667 −1 −266000 667000
A = and A = ,
.333 .266 333000 −835000
then the condition number for A is
6
κ = κ = A A −1 =(1.502)(1168000) = 1, 754, 336 ≈ 1.7 × 10 .
Since the right-hand side of (3.8.8) is only an estimate of the relative error in
the solution, the exact value of κ is not as important as its order of magnitude.
6
Because κ is of order 10 , (3.8.8) holds the possibility that the relative change
(or error) in the solution can be about a million times larger than the relative