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Vector Norms and the related Matrix Norms
13

Vector Norms and the related Matrix Norms

Jan 02, 2016

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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 Presentation
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Page 1: Vector Norms and the related Matrix Norms

Vector Norms and the related Matrix Norms

Page 2: 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

Page 3: Vector Norms and the related Matrix Norms

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

,

Page 4: Vector Norms and the related Matrix Norms

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

Page 5: Vector Norms and the related Matrix Norms

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

Page 6: Vector Norms and the related Matrix Norms

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 , ,

Page 7: Vector Norms and the related Matrix Norms

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

Page 8: Vector Norms and the related Matrix Norms

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

Page 9: Vector Norms and the related Matrix Norms

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

Page 10: Vector Norms and the related Matrix Norms

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

Page 11: Vector Norms and the related Matrix Norms

.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

Page 12: Vector Norms and the related Matrix Norms

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

Page 13: Vector Norms and the related Matrix Norms

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