The Condition Number for a Matrix
James Keesling
1 Condition Numbers
In the section we outline the general idea of a
condition number
. The condition number
is a means of estimating the accuracy of a result in a given calculation. The simplest way
to convey the idea is to do an example. Suppose that numbers on the computer are given
with a certain accuracy
±
. So, a given number
x
is represented roughtly by
x
+
±
. If the
number
x
is represented to machine accuracy on a computer, then the IEEE standards
require that

±


x

≤
2

52
.
Suppose that we are evaluating a function
f
at the point
x
. Let us assume that we
evaluate the function
f
completely accurately with the only error being the representation
of the number
x
. Let
y
=
f
(
x
) be the exact value of
f
at the true value of
x
and let
y
+
δ
=
f
(
x
+
±
). Then we have
f
(
x
+
±
)

f
(
x
)
±
≈
δ
±
≈
f
0
(
x
)
.
We can say that

f
0
(
x
)

is a condition number for the absolute error in the computation
of
f
since

δ
 ≈ 
f
0
(
x
)
·
±

. We can also obtain a condition number for the relative error in
the computation. From the above inequality we have the following.

δ
 ≈ 
f
0
(
x
)
 · 
±


δ


f
(
x
)

≈

f
0
(
x
)


f
(
x
)


x
 ·

±


x

So, the condition number for the absolute error in computing
f
at
x
is

f
0
(
x
)

and the
condition number for the relative error in computing
f
at
x
is

f
0
(
x
)


f
(
x
)

· 
x

.
2 Vector and Matrix Norms
To apply this to matrix computations, we will need a measure of the error. This is done
by means of
vector norms
and
matrix norms
which we now describe.
1
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View Full DocumentLet
V
be an
n

dimensional vector space. Let

x

be a function from
V
to the non
negative real numbers,
 · 
:
V
→
[0
,
∞
). Then

x

is a norm for
x
∈
V
if it satisﬁes the
following axioms.
(1)

x

= 0 if and only if
x
= 0
∈
V
.
(2)

x
+
y
 ≤ 
x

+

y

for all
x,y
∈
V
.
(3)

λ
·
x

=

λ
 · 
x

for all
λ
∈
R
and
x
∈
V
.
The norm on an
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 Summer '08
 Kyung
 Linear Algebra, Statistics, Numerical Analysis, Matrices, Euclidean space, condition number

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