vector norms and the related matrix norms
DESCRIPTION
Vector Norms and the related Matrix Norms. Properties of a Vector Norm: Euclidean Vector Norm: Riemannian metric:. If is a regular vector-norm on the n-dimemsional vector space, and if A is an matrix, we define the related matrix-norm as - PowerPoint PPT PresentationTRANSCRIPT
Vector Norms and the related Matrix Norms
Properties of a Vector Norm:
Euclidean Vector Norm:
Riemannian metric:
)~ ( ||max |~|||max
)( ~~~~) ( ~~)0~ 0~ ( 0~
KKxxxx
yxyx
xx
xxx
KK all for
that such and constants positive are There
inequality triangle the
scalar any for
iffwith
21
1
2~~
n
KKE
xxx
KKS
KKM
ER
j kjkjk
t
R
xx
xx
xxIP
xxPxxPx
P
||~
||max~
~~
)()~,~(~ 21
21
then , If
define definite, positive For
If is a regular vector-norm on the n-dimemsional vector space, and if A is an matrix, we define the related matrix-norm as
Properties of the related matrix-norm:
For some positive constants which are independent of A
x~
nn
|~|max|~|
|~|max||
1|~|0~yA
x
xAA
yx
BAAB
BABA
AA
AA
) (
)0 0 ( 0|A|
)( 1|I|
scalars all for
iffwith
matrix identity the for
||max ||||max ,ji,
,ji,
jiji aAa
,
The Conditional Number of a MatrixIf A is a nonsingular square matrix, we define the conditional
number
Interpretation:Let the unit sphere be mapped by the transformation
into some surface S.The conditional number is the ration of the largest to the
smallest distances from the origin to points on S.Thus,
where are the eigenvalues of A arranged so that
This follows from setting and equal to eigenvectors belonging to and , respectively.
)1~~( ~
~max)( vu
vA
uAAr for
1~ x xAy ~~
)(Ar
1)(
)()( 1
A
AAr
n
n ,,, 21 .21 n
u~ v~
1 n
By the previous definition:
But what is the minimum of ?
we have
Therefore
So,
)1|~|( |~|max|| uuAA for
)max|~|( max ii
jij
ixxa where
1|~| |~| vvA for
|~||||)~(||~| 11 vAAvAAv 1
|~||| 11 vAA
)1|~|( |~|min|| 11 vvAA for
)1|~|~( |~|min
|~|max)( 1 vuAA
vA
uAAr where
Application of Conditional Numbers
Suppose that we are solving , where that data A and are not known exactly. What is the effect of errors and
on the solution?
Let
Assume that A and are nonsingular, and that . Define the error ratios:
bxA~~ b
~
Ab~
bbxxAA~~
)~~)((
AA 0~
b
|~
|
|~
|
|~|
|~|
||
||
b
b
x
x
A
A , ,
We try to estimate as a function of and .
But
Whereas
Therefore,
Multiplying by and division by yield
Hence
then
bbxAbxAxA
xAxAxAxAxxAA~~~)(
~)~)(()~(
~)(~)~)(()~()~~)((
xAbxAxA ~)()~
()~)(()~(
|~||||~||||)~(||)(||)~(|min|)~)((||)~(||)~)(()~(|
11 xAxAxAxAxAxAxAxA
|~||||~
||~)()~
(| xAbxAb
||A-1
|||||~|
|~
||||||| 111 AA
x
bAAA
|||~|
|~|
|~|
|~|
|~
|
|~
|
|~|
|~
|
|~
|
|~
|
|~|
|~
|A
x
xA
x
xA
b
b
x
b
b
b
x
b
|)|||( |||||~|
|~
||| 111 AArrAA
x
bA
)()1( rr
|~||||~
||~||||~|11 xAbxAx||A-
|~| x
|~|
|~|
x
x
|||| 1 AA
Assuming
we find
or
If
then
If
then
1||||||
|||||| 11 AA
A
AAAr
)(1
r
r
)||
||
|~
|
|~
|(
)||||
)((1
)(
|~|
|~|
A
A
b
b
AA
Ar
Ar
x
x
1r
!behaved!- well benot may : )||
||
|~
|
|~
|(
||
||
A
A
b
b
A
A
!behaved!- well: )||
||
|~
|
|~
|(
||||
||
A
A
b
b
AA
A
1)( Ar
Perturbations of the spectrumLet A be an matrix with eigenvalues and with corres
ponding eigenvectors . A small change in the matrix produces changes in the eigenvalues and changes in the eigenvectors.
If are distinct, then are linearly independent and are unique, except for nonzero scalar multiplies
We have
and (1)
In this equation we consider
(2)
If , the ; but the perturbation equation (1) is satisfied by any which is multiple of . To ensure , we shall normalize the perturbated eigenvector by the assumption that,
nn n ,,, 21 nuuu ,,, 21
iA
ju
jj uAu j
j.ju
))(())(( jjjj
jj uuuuAA
known : and , , j juAA
unknown : , j ju
0A ,n),,(jδλ j 21 0 ju ju 00 Au j if
iu
in the expansion
The coefficient of remains equal to 1 when A is replaced by . In other words, we shall require expansions
(3)
The unknowns are now and the coeffs. for .If the components of the matrix are very small, eqn.(1) becomes, to t
he first order,
where the neglected terms, are of second order.Since (4)To compute the unknowns we will use the “principle of bio
rthogonality”:Let be eigenvectors corresponding to the eigenvalues
of an matrix A. assume . Let be eigenvectors corresponding to the eigenvalues of .(Hermitian matrix of A)
Then and
kn
k
jj uuu
1
AA
)0( 1
kkk
n
kjk
j uu for
ju
jA
jj
jj
jj
jjj uuuuAuAAu )()()()( ))(())(( j
jj uuA and
jj
j uAu j
jj
jjj uuuAuA )()()()(
jju and
n ,,, 21 nuuu ,,, 21 ji for ji nvvv ,,, 21
n ,,, 21 HA
0),( jj uu .0),( jiuu ji for
jk kj
nn
.00,
0),(
0),(.
),(),(or ),(),(
),(),( ),(),(
;
.,, 0detdetdet
.detdet
1
iiiki
i
ii
jiji
jij
jii
jj
ijii
H
jj
ijHi
jii
ji
ji
jii
i
nH
it
it
i
ti
Hi
uuuvuv'su
vu
jivu
vuλvuλvλuvuλvuv)(u,A(Au,v)
vλuvAuvuλvAu
vλAvuλAuji
AI-A)(λI-A)(λ)I-A(λ
)AI-λ()I-Aλ(
hence and , yield would tionrepresenta a then , the all to ortogonal were if
because,follows now inequality The
. for that calculated can we , Since
have we, and all forSince
find weproducts, inner Taking
have we,For seigenvalue distinct n the have doesThus,
of conjugate complex the is this But:Proof
To solve eqn.(4) for , we will use the eigenvectors of . By normalization (3), the perturbation is a combination of for . Therefore, .
Now (4) yields,
But
Therefore, since
(5)
To find take the inner product of eqn.(4) with , for :
Since
But the normalization (3) gives
(6)
jj u and nvvv ,,, 21 HA
ju ku jk 0),( jj vu
),)((0),)(()),((
jjj
jjjj
vuvuAvuA
0),()),(()),(( jj
jjHjjj vuvAuvuA 0),( jj vu
njvu
vuAjj
jj
j ,,2,1),(
),)(( ,
ju kv jk
0),(),)(()),(( kjj
kjkj vuvuAvuA
),(),(),()),(( kjk
kk
jkHjkj vuvuvAuvuA
),(),)((),( kjj
kjkjk vuvuAvu
kjvu
vuAkk
kj
kj
jk
,),)((
),)((
),(),( kkjk
kj vuvu
Example: , where . Let , where
and is a small parameter. In this case, we take
for the eigenvectors of A and .Eqn.(5) gives
Eqn.(6) gives
Now Eqn.(3) gives
is the vector whose jth component is 0 and kth component is
ji for 21 BA )( . jibB
)0,,1,,0( col jjj evunj ,,,,1
HA
jjjj
j beBe ),(
)( ),( , kj
beBe
kj
jk
kj
kj
jk
kn
jkk
kjjkj ebu
1
, )(
ju
.)( kj
kjb
)( jk
),,,( 21 nA diag