HP HP-15C Advanced Functions Handbook

Section 4: Using Matrix Operations 101
equality is true. A large condition number makes possible a
relatively large error in the result.
Errors in the data — sometimes very small relative errors — can
cause the solution of an ill-conditioned system to be quite different
from the solution of the original system. In the same way, the
inverse of a perturbed ill-conditioned matrix can be quite different
from the inverse of the unperturbed matrix. But both differences
are bounded by the condition number; they can be relatively large
only if the condition number K( A) is large.
Also, a large condition number K(A) of a nonsingluar matrix A
indicates that the matrix A is relatively close, in norm, to a
singular matrix. That is.
and
where the minimum is taken over all singular matrices S. That is,
if K(A) is large, then the relative difference between A and the
closest singular matrix S is small. If the norm of A'1 is large, the
difference between A and the closest singular matrix S is small.
For example, let
Then
.9999999999
-9,999,999,999 1010
1010 -1010
and ||A 1|| = 2 X 1010. Therefore, there should exist a perturbation
AA with || AA|| = 5 X 10"11 that makes A + AA singular. Indeed, if
AA =
with ||AA|| = 5 X 10"11, then
A + AA =
0 -5X10"11
0 5XKT11
1 .99999999995
1 .99999999995

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