HP 15c - Manual (Page 85)

Section 4: Using Matrix Operations 85
that X and B are nonzero vectors satisfying AX = B for some square matrix A. Suppose A is
perturbed by ΔA and we compute B + ΔB = (A + ΔA)X. Then
 
  ) (A
A
ΔA
B
ΔB
K

,
with equality for some perturbation ΔA. This measures how much the relative uncertainty in
A can be magnified when propagated into the product.
The condition number also measures how much larger in norm the relative uncertainty of the
solution to a system can be compared to that of the stored data. Suppose again that X and B
are nonzero vectors satisfying AX = B for some matrix A. Suppose now that matrix B is
perturbed (by rounding errors, for example) by an amount ΔB. Let X + ΔX satisfy
A(X + ΔX) = B +ΔB. Then
 
  ) (A
B
ΔB
X
ΔX
K

with equality for some perturbation ΔB.
Suppose instead that matrix A is perturbed by ΔA. Let X + ΔX satisfy (A + ΔA)(X + ΔX) =
B. If d(A, ΔA) = K(A)||ΔA|| /||A|| < 1, then
 
 
) ( 1
) (
ΔA
A,
A
A
ΔA
X ΔX
d
K


.
Similarly, if A
−1
+ Z is the inverse of the perturbed matrix A + ΔA, then
 
 
) ( 1
) (
ΔA
A,
A
A
ΔA
A Z
1
d
K



.
Moreover, certain perturbations ΔA cause the inequalities to become equalities.
All of the preceding relationships show how the relative error of the result is related to the
relative error of matrix A via the condition number K(A). For each inequality, there are
matrices for which 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 nonsingular matrix A indicates that the matrix A is
relatively close, in norm, to a singular matrix. That is