Chemistry Reference
In-Depth Information
There are many particular vector norms which are used in practical applications.
A large class of matrix norms can be generated from vector norms in the following
way. Let
kk
be a given vector norm in
n
(see Appendix A), and for any
A
2 R
n
n
R
define
k
Av
k
k
v
k
k
A
kD
max
v
D
max
k
v
kD1
k
Av
k
,
¤
0
where the numerator and the denominator use the same given vector norm. It is easy
to prove that matrix norms generated by vectors norms always satisfy conditions
(1)-(3) and in addition, for all
A
and
B
2 R
nn
,
4.
k
AB
kk
A
kk
B
k
.
Furthermore it is an additional important fact, that for all
v
2 R
n
and
A
2 R
n
n
,
5.
k
Av
kk
A
kk
v
k
; if the matrix norm
k
A
k
is generated from the vector norm
which is used to compute both
k
v
k
and
k
Av
k
.
If property (5) holds for a given vector norm and a particular matrix norm, then we
say that the two norms are
compatible
.
The matrix norms generated from the vector norms
k
v
k
1
;
k
v
k
1
and
k
v
k
2
are
given as follows. Let a
ij
denote the .i;j/ elements of matrix
A
,then
8
<
9
=
X
n
k
A
k
1
D
max
i
1
j
a
ij
j
.row norm/
:
;
j
D
(
n
)
X
i D1
j
a
ij
j
k
A
k
1
D
max
j
.column norm/
and
r
max
i
i
.
A
T
A
/ Euclidean norm/
k
A
k
2
D
where
i
.
A
T
A
/.i
D
1;2;:::;n/are the eigenvalues of the product
A
T
A
,and
A
T
is the transpose of
A
. It can be proved that all eigenvalues of
A
T
A
are real and
nonnegative.
Notice that conditions (1)-(3) for matrix norms do not imply that there is a
vector norm which is compatible with it. For example, consider the matrix norm
k
A
kD
1
2
k
A
k
1
,then
k
I
kD
1
2
, so with
a
¤
0
and any vector norm,
k
Ix
kDk
x
k
>
1
2
k
x
kDk
I
kk
x
k
:
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