eigen value, eigen vector and its application

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TERM PAPER

ENGINEERING MATHEMATICS

(MTH101)

Topic

::

EIG

EIGEN

EN VAL

VALUES

UES AN

AND

D EI

EIGE

GEN

N

V

VE

EC

CT

TO

OR

RS

S

A

AN

ND

D

IIT

TS

S

APPLICATIONS

DOA:

DOA: 14

14 Sep

Sep 2010

2010

DOR:

DOR: 19

19 Oct

Oct 2010

2010

DOS: 16 Nov 2010

Submitted to:

Submitted by:

Ms.

Ms. Priyanka

Priyanka Singh

Singh

Mr.

Amandeep Singh Khera

Deptt. Of Mathematics

Roll. No.

Roll. No. RK6005A19

RK6005A19

Reg.No.

11000597

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Class

K6005

Class

K6005

ACKNOWLEDGEMENT

IIt

t aacck

kn

no

ow

wlleed

dg

gees

s aalll

l tth

he

e cco

on

nttrriib

bu

utto

orrs

s iin

nv

vo

ollv

veed

d iin

n tth

hee

preparation of this project. Including me, there is a hand of

m

my

y te

teaach

cher

ers,

s, so

som

me

e b

bo

oo

oks

ks an

and

d in

inte

terrn

net

et.

. I

I eexp

xpre

ress

ss m

mo

ost

st

gratitude to my subject teacher, who guided me in the right

direction. The guidelines provided by her helped me a lot in

completing the assignment.

The books and websites I consulted helped me to describe

ea

each

ch an

and

d ev

ever

ery

y po

poin

int

t me

ment

ntio

ione

ned

d in

in th

this

is pr

proj

ojec

ect.

t. He

Help

lp of

of

or

orig

igin

inal

al cr

crea

eati

tivi

vity

ty an

and

d il

illu

lust

stra

rati

tion

on ha

had

d ta

take

ken

n an

and

d I

I ha

have

ve

explained each and every aspect of the project precisely.

At last it acknowledges all the members who are involved in

the preparation of this project.

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Thanks

AMANDEEP SINGH

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ABSTRACT

A

Aft

fter

er g

goi

oing

ng th

thro

roug

ugh

h th

thiis

s p

pap

aper

er on

one

e w

wil

ill

l co

com

me

e

across what is eigenvector and eigenvalue. What

is the importance of eigen value and eigenvector

in ou

in our day to d

r day to day

ay lif

life and h

e and hist

istory o

ory of eig

f eigenv

envalu

alue

e

and

and eig

eigenv

envect

ector

or and

and its

its var

variou

ious

s app

applic

licati

ations

ons in

in

sc

schr

hrod

odin

inge

ger

r eq

equa

uati

tion

on,g

,geo

eolo

logy

gy an

and

d ei

eige

gen

n fa

face

ces

s

etc.

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TABLE OF CONTENT

1.

INTRODUCTION TO EIGEN VALUE

AND EIGEN VECTOR

2

2. .

H

HIIS

ST

TO

OR

RY

Y

3

3. .

A

AP

PP

PL

LIIC

CA

AT

TIIO

ON

N O

OF

F E

EIIG

GE

EN

N V

VA

AL

LU

UE

E

AND EIGEN VECTOR

3.1 3.1 SCHRODINGER EQUATION

SCHRODINGER EQUATION

3.2 3.2 MOLECULAR ORBITAL

MOLECULAR ORBITAL

3.3 3.3 GEOLOGY AND

GEOLOGY AND

GLACIOLOGY

3.4 3.4 FACTOR ANALYSIS

FACTOR ANALYSIS

3.5 3.5 VIBRATION ANALYSIS

VIBRATION ANALYSIS

3.6 3.6 EIGEN FACES

EIGEN FACES

3.7 3.7 TENSOR OF INERTIA

TENSOR OF INERTIA

3.8 3.8 STRESS TENSOR

STRESS TENSOR

3.9 3.9 EIGEN VALUE OF A GRAPH

EIGEN VALUE OF A GRAPH

4.

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INTRODUCTION TO EIGEN VALUE

AND EIGEN VECTOR

They are derived from the German word "eigen" which

means "proper" or "characteristic." An eigenvalue of a

square matrix is a scalar that is usually represented by

the Gr

the Greek le

eek lette

tter

r (pr

(prono

onounc

unced lam

ed lambda

bda). As you mi

). As you might

ght

su

susp

spec

ect,

t, an

an ei

eige

genv

nvec

ecto

tor

r is

is a

a ve

vect

ctor

or.

. Mo

More

reov

over

er,

, we

we

require that an eigenvector be a non-zero vector, in

other words, an eigenvector can not be the zero vector.

We will denote an eigenvector by the small letter

We will denote an eigenvector by the small letter x

x . All

. All

ei

eige

genv

nval

alue

ues

s an

and

d ei

eige

genv

nvec

ecto

tors

rs sa

sati

tisf

sfy

y th

the

e eq

equa

uati

tion

on

for a given square matrix,

for a given square matrix, A

A..

Con

Consid

sider

er the

the squ

square

are ma

matri

trix

x A

A . W

. We s

e sa

ay t

y th

ha

at

t iis a

s an

n

eigenvalue

eigenvalue of

of A

A if there exists a non-zero vector

if there exists a non-zero vector x

x

s

su

uc

ch

h tth

ha

at

t

.

. IIn

n tth

hiis

s c

ca

as

se

e,, x

x iis

s c

ca

alllle

ed

d a

an

n

eigenvector

eigenvector (c

(cor

orre

resp

spon

ondi

ding

ng to

to ),

), an

and

d th

the

e pa

pair

ir (( ,, x

x )

) is

is

called an

called an eigenpair

eigenpair for

for A

A..

Let's look at an example of an eigenvalue and eigenvector. If you Let's look at an example of an eigenvalue and eigenvector. If you

wer

were e askasked ed if if is is an an eigeigenvenvectector or corcorresresponpondinding g to to thethe eigenvalue

eigenvalue for, for, you you could could find find out out by by substitutingsubstituting x x, and, and A A intointo the equation

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Th

Ther

eref

efor

ore,

e, an

and

d x

x ar

are

e an

an ei

eige

genv

nval

alue

ue an

and

d an

an ei

eige

genv

nvec

ecto

tor,

r,

respectively, for

respectively, for A

A..

HISTORY

E

Eiiggeennvvaalluuees s aarre e oofftteen n iinnttrroodduucceed d iin n tthhe e ccoonntteexxt t oof f linear linear algebra

algebra or or matrmatrix ix theortheoryy. . HiHiststororicicalallyly, , hohowewevever, r, ththey ey ararosose e in in ththee study of

study of quadratic formsquadratic forms andand differentidifferential al equationsequations.. Euler

Euler hahad d alalso so ststududieied d ththe e rorotatatitiononal al mmototioion n of of aa rigrigid id bodbodyy andand di

discscovoverered ed ththe e imimpoportrtanance ce of of ththee pr princincipaipal l axeaxess. . AAs s LLaaggrraannggee re

realalizizeded, , ththe e prprinincicipapal l axaxes es arare e ththe e eieigegenvnvecectotors rs of of ththe e ininerertitiaa matrix. In the early 19th century,

matrix. In the early 19th century, CauchyCauchy saw how their work couldsaw how their work could be used to classify the

be used to classify the quadric surfacesquadric surfaces, and generalized it to arbitrary, and generalized it to arbitrary d

diimmeennssiioonnss. . CCaauucchhy y aallsso o ccooiinneed d tthhe e tteerrmmracineracine caractéristique

caractéristique (characteristi(characteristic root) for what c root) for what is now calledis now called eigenvalueeigenvalue;; his term survives in

his term survives in characteristic equationcharacteristic equation.. Fourier

Fourier used the work of Laplace and Lagrange to solve theused the work of Laplace and Lagrange to solve the heatheat equation

equation byby separation of variablesseparation of variables in his famous 1822 book in his famous 1822 book ThéorieThéorie analytique de la chaleur

analytique de la chaleur .. SturmSturm developed Fourier's ideas further anddeveloped Fourier's ideas further and he brought them to the attention of Cauchy, who combined them with he brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that

his own ideas and arrived at the fact that symmetric matrices have realsymmetric matrices have real eigen

eigenvaluevalues. This. This s was was extenextended ded byby HermiteHermite in 1855 to what are nowin 1855 to what are now called

called HermHermitiaitian n matrmatricesices. . ArArouound nd ththe e sasame me titimeme,, BrioschiBrioschi provedproved tthhaat t tthhe e eeiiggeennvvaalluuees s oof f ororththogogononal al mamatrtriciceses lliie e oon n tthhee unitunit circle

circle, and, andClebschClebsch found the corresponding result for found the corresponding result for skew-symmetricskew-symmetric matrices

matrices. Finally,. Finally, WeierstrassWeierstrass ccllaarriiffiieed d aan n iimmppoorrttaannt t aassppeecct t iinn the

the stastabilbility ity thetheoryory ststararteted d by by LaLaplplacace e by by rerealalizizining g ththatatdefectivedefective matrices

matrices can cause instability.can cause instability. In

In the the meameantintime,me, LiouvilleLiouville stustudiedied d eigeigenvenvalualue e proprobleblems ms simsimilailar r toto those of Sturm; the discipline that grew out of their work is now those of Sturm; the discipline that grew out of their work is now

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called

called SturmSturm-Liouv-Liouville ille theotheoryry.. SchwarzSchwarz ststududieied d ththe e fifirsrst t eieigegenvnvalalueue of

of Laplace's equationLaplace's equation on general domains towards the end of the 19thon general domains towards the end of the 19th century, while

century, while PoincaréPoincaré studiedstudied Poisson's equationPoisson's equation a few years later.a few years later. At

At tthe he ststarart t of of tthe he 220t0th h ccenentturury,y, HilbertHilbert stustudiedied d the the eigeigenvenvalualueses of

of integral operatorsintegral operators by viewing the operators as infinite matrices. Heby viewing the operators as infinite matrices. He was the first to use the

was the first to use the GermanGerman wordword eigeneigen to denote eigenvalues andto denote eigenvalues and eigenvectors in 1904, though he may have been following a related eigenvectors in 1904, though he may have been following a related usage by

usage by HelmholtzHelmholtz. "Eigen" can be translated as "own", "peculiar . "Eigen" can be translated as "own", "peculiar to", "characteristic", or "individual" — emphasizing how important to", "characteristic", or "individual" — emphasizing how important eeiiggeennvvaalluuees s aarre e tto o ddeeffiinniinng g tthhe e uunniiqquue e nnaattuurre e oof f a a ssppeecciiffiicc tr

transansforformatmationion. . For For somsome e titime, me, the the stastandandard rd terterm m in in EngEnglilish sh waswas "proper value", but the more distinctive term "eigenvalue" is standard "proper value", but the more distinctive term "eigenvalue" is standard today.

today. Th

The e fifirsrst t nunumemeriricacal l alalgogoririththm m fofor r cocompmpututining g eieigegenvnvalalueues s anandd eigenvectors appeared in 1929, when

eigenvectors appeared in 1929, when Von MisesVon Mises published thepublished the power power method

method. One of the most popular methods today, the. One of the most popular methods today, the QR algorithmQR algorithm,, w

waas s pprrooppoosseed d iinnddeeppeennddeennttlly y bbyy JJoohhn n GG..FF. . FFrraanncciiss andand VeraVera Kublanovskaya

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APPLICATIONS OF EIGEN VALUES

AND EIGEN VECTORS

Schrödinger Equation

An

An examexample ple of of an an eigeeigenvalunvalue e equatequation ion wherwhere e the the trantransforsformatimation T ison T is rreepprreesseenntteed d iin n tteerrmms s oof f a a ddiiffffeerreentntiiaal l ooppeerraattoor r iis s tthhe e ttiim mee--independent

independent Schrödinger equationSchrödinger equation inin quantum mechanicsquantum mechanics::

w

whheerre e HH, , tthhee HamiltonianHamiltonian, , iis s a a sseeccoonndd--oorrddeer r differentialdifferential operator

operator anand d ψEψE, , ththee wavefunctionwavefunction, , iis s oonne e oof f iitts s eeiiggeennffuunnccttiioonnss corresponding to the eigenvalue E, interpreted as its

corresponding to the eigenvalue E, interpreted as its energyenergy.. Ho

Howewevever, r, in in ththe e cacase se whwherere e onone e is is ininteterereststed ed ononly ly in in ththee bound bound state

state solutions of the Schrödinger equation, one looks for ψE withinsolutions of the Schrödinger equation, one looks for ψE within the space of

the space of square integrablesquare integrable functions. Since this space is afunctions. Since this space is a HilbertHilbert space

space with a well-definedwith a well-defined scalar productscalar product, one can introduce a, one can introduce a basis basis set

set in which ψE and H can be represented as a one-dimensional arrayin which ψE and H can be represented as a one-dimensional array aannd d a a mmaattrriix x rreessppeeccttiivveellyy. . TThhiis s aalllloowws s oonne e tto o rreepprreesseennt t tthhee Schrödinger equation in a matrix form. (Fig. 8 presents the lowest Schrödinger equation in a matrix form. (Fig. 8 presents the lowest eigenfunctions of the

eigenfunctions of the Hydrogen atomHydrogen atom Hamiltonian.)Hamiltonian.) The

The DiraDirac c notanotationtion is is ofofteten n usused ed in in ththis is cocontntexext. t. A A vevectctoror, , whwhicichh re

reprpresesenents ts a a ststatate e of of ththe e sysyststemem, , in in ththe e HiHilblberert t spspacace e of of sqsquauarere in

intetegrgrabable fle fununctctioions ins is res reprpresesenenteted by d by . I. In thn this nis nototatatioion, tn, thehe Schrödinger equation is:

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wh

wherere e is an is an eieigegensnstatate te of of H. IH. It is at is a selself f adjadjoinoint t opeoperatrator or , , tthhee infinite dimensional analog of Hermitian matrices (see

infinite dimensional analog of Hermitian matrices (see ObservableObservable).). As

As in in the the matrmatrix ix case, case, in in the the equatequation ion abovabove e is is underunderstood stood toto be

be the the vector vector obtained obtained by by application application of of the the transformatiotransformation n H H to to ..

F

Fiig

g.

. 8

8.. The

The wavefunctions

wavefunctions as

asso

soci

ciat

ated

ed wi

with

th th

the

ebound

bound

states

states of an

of an electron

electron in

in a

a hydrogen atom

hydrogen atom can be seen

can be seen

a

as

s

tth

he

e

eiig

ge

en

nv

ve

ec

ctto

orrs

s

of

o

f

tth

he

e h

hy

yd

drro

og

ge

en

n

a

atto

om

m

Hamiltonian

Hamiltonian a

as

s w

we

elll

l a

as

s o

of

f tth

he

e ang

angul

ular

ar mo

momen

mentum

tum

operator

operator .

. T

Th

he

ey

y a

arre

e a

as

ss

so

oc

ciia

atte

ed

d w

wiitth

h e

eiig

ge

en

nv

va

allu

ue

es

s

iin

ntte

errp

prre

ette

ed

d

a

as

s

tth

he

eiir

r

e

en

ne

errg

giie

es

s

((iin

nc

crre

ea

as

siin

ng

g

d

do

ow

wn

nw

wa

arrd

d::n

n=

=1

1,,2

2,,3

3,,...)

)

a

an

nd

d angular

angular

momentum

momentum ((iin

nc

crre

ea

as

siin

ng

g

a

ac

crro

os

ss

s::s

s,

, p

p,

, d

d,,...)).

.

T

Th

he

e

illustration shows the square of the absolute value of

th

the

e wa

wave

vefu

func

ncti

tion

ons.

s. B

Bri

righ

ghte

ter

r ar

are

eas

as c

co

orr

rre

esp

spon

ond

d to

to

higher

higher pro

probabil

bability

ity densi

density

ty fo

for

r a

a po

posi

siti

tion

onmeasurement

measurement ..

T

Th

he

e c

ce

en

ntte

er

r o

of

f e

ea

ac

ch

h ffiig

gu

urre

e iis

s tth

he

e at

atom

omic

ic nu

nucl

cleu

eus

s,,

a

a proton

proton..

Molecular Orbitals

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In

In quantquantum um mechmechanicsanics, , aannd d iin n ppaarrttiiccuullaar r iinn atomicatomic andand molecular molecular physics

physics, , wwiitthhiin n tthhee Hartree-Fock Hartree-Fock ththeoeoryry, , ththee atomicatomic andand molecular molecular orbitals

orbitals can be defined by the eigenvectors of thecan be defined by the eigenvectors of the Fock operator Fock operator . The. The ccoorrrreessppoonnddiinng g eeiiggeennvvaalluuees s aarre e iinntteerrpprreetteed d aass ionizationionization potentials

potentialsviavia Koopmans' theoremKoopmans' theorem. In this case, the term eigenvector is. In this case, the term eigenvector is used in a somewhat more general meaning, since the Fock operator is used in a somewhat more general meaning, since the Fock operator is exp

expliclicitlitly y depdependendent ent on on the the orborbitaitals ls and and thetheir ir eigeigenvenvalualues. es. If If oneone wants to underline this aspect one speaks of nonlinear eigenvalue wants to underline this aspect one speaks of nonlinear eigenvalue problem. Such equations are usually solved by an

problem. Such equations are usually solved by an iterationiteration procedure,procedure, called in this case

called in this case self-consistent fieldself-consistent field method. Inmethod. In quantum chemistryquantum chemistry,, o

onne e oofftteen n rreepprreesseenntts s tthhe e HHaarrttrreeee--FFoocck k eeqquuaattiioon n iin n a a nnoon n--orthogonal

orthogonal basis set basis set. This particular representation is a generalized. This particular representation is a generalized eigenvalue problem called

eigenvalue problem called Roothaan equationsRoothaan equations..

Geology And Glaciology

In

In geologygeology, especially in the study of , especially in the study of glacial tillglacial till, eigenvectors and, eigenvectors and eigenvalues are used as a method by which a mass of information of a eigenvalues are used as a method by which a mass of information of a clast fabric's constituents' orientation and dip can be summarized in a clast fabric's constituents' orientation and dip can be summarized in a 3-D space by six numbers. In the field, a geologist may collect such 3-D space by six numbers. In the field, a geologist may collect such data for hundreds or thousands of

data for hundreds or thousands of clastsclasts in a soil sample, which canin a soil sample, which can only be compared graphically such as in a Tri-Plot (Sneed and Folk) only be compared graphically such as in a Tri-Plot (Sneed and Folk) di

diagagraram m or or as as a a StSterereoeonenet t on on a a WuWulflff f NeNet t . . ThThe e ououtptput ut fofor r ththee orientation tensor is in the three orthogonal (perpendicular) axes of orientation tensor is in the three orthogonal (perpendicular) axes of space. Eigenvectors output from programs such as Stereo32 are in the space. Eigenvectors output from programs such as Stereo32 are in the ord

order er E1 E1 ≥ ≥ E2 E2 ≥ ≥ E3, E3, witwith h E1 E1 beibeing ng the the priprimarmary y orioriententatiation on of of claclastst orientation/dip

orientation/dip, , E2 E2 being the secondary and being the secondary and E3 E3 being the tertiary, inbeing the tertiary, in terms of strength.

terms of strength.

The clast orientation is defined as the eigenvector, on a compass rose The clast orientation is defined as the eigenvector, on a compass rose of 360°. Dip is measured as the eigenvalue, the modulus of the tensor: of 360°. Dip is measured as the eigenvalue, the modulus of the tensor:

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this is valued from 0° (no dip) to 90° (vertical). The relative values this is valued from 0° (no dip) to 90° (vertical). The relative values of E1, E2, and E3 are dictated by the nature of the sediment's fabric. of E1, E2, and E3 are dictated by the nature of the sediment's fabric. If E1

If E1 = E2 = E2 = E3, the = E3, the fabrfabric ic is is said to said to be be isotrisotropicopic. . If E1 = If E1 = E2 > EE2 > E3the3the fabric is planar. If E1 > E2 > E3 the fabric is linear. See 'A Practical fabric is planar. If E1 > E2 > E3 the fabric is linear. See 'A Practical Guide to the Study of Glacial Sediments' by Benn & Evans, 2004 . Guide to the Study of Glacial Sediments' by Benn & Evans, 2004 .

Factor analysis

In

In ffaaccttoor r aannaallyyssiiss, , tthhe e eeiiggeennvveeccttoorrss of of aa covariancecovariance

matrix

matrix or or correlation matrixcorrelation matrix correspond tocorrespond to factorsfactors, and eigenvalues to, and eigenvalues to tthhe e vvaarriiaanncce e eexxppllaaiinneed d bby y tthheesse e ffaaccttoorrss. . FFaaccttoor r aannaallyyssiis s iiss aastatisticalstatistical tteecchhnniiqquue e uusseed d iin n tthhee ssoocciiaal l sscciieenncceess andand in

in marketingmarketing,, pro product duct managmanagementement,, operoperationations s reseresearcharch, , aannd d ootthheer r applied sciences that deal with large quantities of data. The objective applied sciences that deal with large quantities of data. The objective iis s tto o eexxppllaaiin n mmoosst t oof f tthhe e ccoovvaarriiaabbiilliitty y aammoonng g a a nnuummbbeer r oof f observable

observable ranrandom dom varvariabiablesles iin n tteerrmms s oof f a a ssmmaalllleer r nnuummbbeer r oof f unobservable latent variables called factors. The observable random unobservable latent variables called factors. The observable random va

variriabableles s arare e momodedeleled d asas linelinear ar combcombinatinationsions of of ththe e fafactctorors, s, plplusus unique variance terms. Eigenvalues are used in analysis used by unique variance terms. Eigenvalues are used in analysis used by Q-methodology software; factors with eigenvalues greater than 1.00 are methodology software; factors with eigenvalues greater than 1.00 are co

consnsididerered ed sisigngnifificicanant, t, exexplplaiainining ng an an imimpoportrtanant t amamouount nt of of ththee variability in the data, while eigenvalues less than 1.00 are considered variability in the data, while eigenvalues less than 1.00 are considered too weak, not explaining a

too weak, not explaining a significant portion of the data variability.significant portion of the data variability.

Vibration analysis

Ei

Eigegenvnvalalue ue prproboblelems ms ococcucur r nanatuturaralllly y in in ththe e vivibrbratatioion n ananalalysysis is of of mechanical structures with many

mechanical structures with many degrees of freedomdegrees of freedom. The eigenvalues. The eigenvalues are used to determine the natural frequencies of vibration, and the are used to determine the natural frequencies of vibration, and the eigenvectors determine the shapes of these vibrational modes. The eigenvectors determine the shapes of these vibrational modes. The

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orthogonality properties of the eigenvectors allows decoupling of the orthogonality properties of the eigenvectors allows decoupling of the differential equations so that the system can be represented as linear differential equations so that the system can be represented as linear summation of the eigenvectors. The eigenvalue problem of complex summation of the eigenvectors. The eigenvalue problem of complex structures is often solved using

structures is often solved using finite element analysisfinite element analysis..

Eigen Faces

Fig. shows eigen faces as eigen vectors Fig. shows eigen faces as eigen vectors

In

In image processingimage processing, processed images of , processed images of facesfaces can be seen as vectorscan be seen as vectors whose components are the

whose components are the brightnesses brightnesses of eachof each pixel pixel. The dimension. The dimension of this vector space is the number of pixels. The eigenvectors of of this vector space is the number of pixels. The eigenvectors of the

the covariance matrixcovariance matrix associated to a large set of normalized picturesassociated to a large set of normalized pictures o

of f ffaaccees s aarre e ccaalllleedd eigenfaceseigenfaces; ; tthhiis s iis s aan n eexxaammpplle e oof f principal principal

components analysis

components analysis. They are very useful for expressing any face. They are very useful for expressing any face iimmaagge e aas s aa linlinear ear comcombinbinatiationonoof f ssoomme e oof f tthheemm. . IIn n tthhee facialfacial

recognition

recognition brbrananch ch of of biometrics biometrics, , eieigegennffacaces es pprorovivide de a a mmeaeans ns oof f applying

applying ddatata a cocommprpreessssiionon tto o ffaaccees s ffoor r identificationidentification purposes. purposes. Research related to eigen vision systems determining hand gestures Research related to eigen vision systems determining hand gestures has also been made.

has also been made.

Similar to this concept, eigenvoices represent the general direction of Similar to this concept, eigenvoices represent the general direction of variability in human pronunciations of a particular utterance, such as a variability in human pronunciations of a particular utterance, such as a w

woorrd d iin n a a llaanngguuaaggee. . BBaasseed d oon n a a lliinneeaar r ccoommbbiinnaattiioon n oof f ssuucchh eeiiggeennvvooiicceess, , a a nneew w vvooiicce e pprroonnuunncciiaattiioon n oof f tthhe e wwoorrd d ccaan n bbee

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con

constrstructucted. ed. TheThese se conconcepcepts ts havhave e beebeen n foufound nd useuseful ful in in autautomaomatiticc speech recognition systems, for speaker adaptation.

speech recognition systems, for speaker adaptation.

Tensor Of Inertia

In

In mechanicsmechanics, , tthhe e eeiiggeennvveeccttoorrs s oof f tthhee iinenerrttiia a ttenensosor r definedefine the

the pr princincipipal al axeaxessof of aa ririgigid d bobodydy. . TThhee tensor tensor of of inertiainertia iis s a a kkeeyy quantity required in order to determine the rotation of a rigid body quantity required in order to determine the rotation of a rigid body around its

around its center of masscenter of mass..

Stress Tensor

In

In solisolid d mechmechanicsanics, the, the strstress ess tentensor sor is is sysymmmmetetrric ic anand d so so cacan n bebe de

decocompmpososed ed ininto to aa diagonaldiagonal tetensnsor or wiwith th ththe e eieigegenvnvalalueues s on on ththee diagonal and eigenvectors as a basis. Because it is diagonal, in this diagonal and eigenvectors as a basis. Because it is diagonal, in this o

orriieennttaattiioonn, , tthhe e ssttrreesss s tteennssoor r hhaas s nnoo shear shear cocompmpononenentsts; ; ththee components it does have are

components it does have are the principal components.the principal components.

Eigenvalues Of A Graph

In

In spectral graph theoryspectral graph theory, an eigenvalue of a, an eigenvalue of a graphgraph is defined as anis defined as an eigenvalue of the graph's

eigenvalue of the graph's adjacency matrixadjacency matrix A, or (increasingly) of theA, or (increasingly) of the graph's

graph's LaplacianLaplacian mamatrtrixix, , whwhicich h is is eieithther er T−T−A A or or I−I−T1T1/2/2AT AT −1−1/2/2,, where T is a diagonal matrix holding the degree of each vertex, and where T is a diagonal matrix holding the degree of each vertex, and in T −1/2, 0 is substituted for 0−1/2. The kth principal eigenvector of in T −1/2, 0 is substituted for 0−1/2. The kth principal eigenvector of a graph is defined as either the eigenvector corresponding to the kth a graph is defined as either the eigenvector corresponding to the kth largest eigenvalue of A, or the eigenvector corresponding to the kth largest eigenvalue of A, or the eigenvector corresponding to the kth smallest eigenvalue of the Laplacian. The first principal eigenvector smallest eigenvalue of the Laplacian. The first principal eigenvector of the graph is

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Th

The e prprinincicipapal l eieigegenvnvecectotor r is is usused ed to to memeasasurure e ththee centralitycentrality of of ititss vertices. An example is

vertices. An example is GoogleGoogle's's PageRank PageRank algorithm. The principalalgorithm. The principal eigenvector of a modified

eigenvector of a modified adjacency matrixadjacency matrix of the World Wide Webof the World Wide Web graph gives the page ranks as

graph gives the page ranks as its components. This vector correspondsits components. This vector corresponds to the

to the stationary distributionstationary distribution of theof the Markov chainMarkov chain represented by therepresented by the row

row-n-normormalialized zed adjadjaceacency ncy matmatrixrix; ; howhoweveever, r, the the adjadjaceacency ncy matmatririxx must first be modified to ensure a stationary distribut

must first be modified to ensure a stationary distribution exists. Theion exists. The second principal eigenvector can be used to partition the graph into second principal eigenvector can be used to partition the graph into clusters, via

clusters, viaspectral clusteringspectral clustering. Other methods are also available for . Other methods are also available for clustering.

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BIBLOGRAPHY BIBLOGRAPHY

1.

1. en.wikipedia.org/.../Eigenvalue,_eigenvector_and_eigenspaceen.wikipedia.org/.../Eigenvalue,_eigenvector_and_eigenspace 2.

2. mathworld.wolfram.com › ... ›mathworld.wolfram.com › ... › MatricesMatrices ›› Matrix EigenvaluesMatrix Eigenvalues 3. 3. www.sosmath.com/matrix/eigen0/eigen0.htmlwww.sosmath.com/matrix/eigen0/eigen0.html 4. 4. www.eigenvalue.comwww.eigenvalue.com 5. 5. planetmath.org/encyclopedia/Eigenvalue.htmlplanetmath.org/encyclopedia/Eigenvalue.html 6.

6. higher engineering mathematicshigher engineering mathematics 7.