線性代數(shù)難點解析（Analysisofdifficultiesinlinearalgebra）

線性代數(shù)難點解析(Analysisofdifficultiesinlinearalgebra)

Ananalysisofdifficultproblemsinlinearalgebra.Txt

Chapterdeterminant

I.emphasis

1.Understanding:thedefinitionofadeterminant,acofactor,

analgebraiccofactor.

2,grasp:determinantofthebasicnatureandinference.

3,theuseof:theuseofdeterminantpropertiesandcalculation

methodstocalculatethedeterminant,usingtheClemrulefor

solvingequations.

Two,difficulties

Theapplicationofdeterminantinthesolutionoflinear

equations,theinverseofmatrices,the1inearcorrelationof

vectorsandtheeigenvaluesofmatrices.

Threeimportantformula

1,ifAisamatrixofN,then,kA/=kn/A/

2,ifAandBarenordermatrix,is/AB/A/B/=,,,

3,ifAisamatrixofN,then,A*//A/n-1=

IfAisninvertiblematrix,then/A-l/A/-1/=

4,ifAisnordersquare,lambdaI(i=l,2,...N,A)isthe

characteristicvalueofA/PIlambda=I

Four,questionsandSolutions

1,thepropositionabouttheconceptandpropertyof

determinant

2,thecalculationofthedeterminant(method)

1)usedefined

2)reducetheorderofadeterminantaccordingtoarow(column)

3)thenatureofthedeterminant

Rows(columns)addedtothesamerow(column)tobeappliedto

theequalityoftheelementsofeachcolumn(row).

Doubleorminusthesameline(column)ofeachrow(column),

reducethedeterminant,orturnitintotheupper(lower)

triangledeterminant.

Third,successive(column)additionandsubtraction,and

simplifieddeterminant.

Breakthedeterminantintothesumanddifferenceofseveral

determinants.

4)recursivemethodisapplicabletothedeterminantwith

strongregularityandzeroelements

5)mathematicalinduction,moreusedtoprove

3,useClem'slawtosolvelinearequations

IfD=A//=0,thenAx=bhasauniquesolution,i.e.

X1=D1/D,x2=,D2/D,...Xn=Dn/D

WhereDjistochangethecoefficientsofXJintoconstantsin

D.

Note:theClemlawappliesonlytoequationswhosenumberof

equationsisequaltothenumberofunknowns.

4,usingthecoefficientdeterminantsolutionoftheproblem

ofdiscriminationofAequations

1)when,A/=0,Ax=0homogeneousequationswithnonzero

solution;non-homogeneousequationsAx=Bisnottheonly

solution(mayhavenosolution,mayalsohaveinfinitelymany

solutions)

2)when,A/=0,theequationAx=0onlyzerosolution;

non-homogeneousequationAx=Bistheonlysolution,this

solutioncanbecalculatedbytheClemrule

Secondchaptermatrix

I.emphasis

1.Understanding:thedefinitionandpropertiesofmatrices,

severalspecialmatrices(zeromatrix,upper(lower)

triangularmatrix,symmetricmatrix,diagonalmatrix,inverse

matrix,orthogonalmatrix,adjointmatrix,partitionedmatrix)

2,master:

1)matrixofvariousoperationsandrulesofoperation

2)themethodofjudginginvertiblematrixandinvertingmatrix

3)elementarytransformationmethodofmatrix

Two,difficulties

1、theelementarytransformationofinversematrixofmatrix

2,therelationbetweenelementarytransformationand

elementarymatrix

Three,theimportantformulaanddifficultyanalysis

1,linearoperation

1)theexchangelawisgenerallynotestablished,namelyABand

BA

2)somealgebraicidentitiescannotbeapplieddirectly,such

asA,BandC,allofwhicharenordermatrices

(A+B)2=A2+AB+BA+B2=A2+2AB+B2

(AB)2=(AB)(AB=A2B2)

(AB)k=AkBk

(A+B)(A-B=A2-B2)

AllofthesearesetuponlywhenAandBareexchanged,i.e.,

AB=BA.

3)A=0orB=0cannotbederivedfromAB=0

4)theAB=ACcannotbereachedbyB=C

5)A=IorA=0cannotbederivedfromA2=A

6)theA2=0cannotbereachedbyA=0

7)thedifferencebetweennumbermultiplicationmatrixand

numbermultiplicationdeterminant

2andinversematrix

1)(A-1)-1=A

2)(kA)-1=(1/k)A-1(k=0).

3)(AB)-1=B-1A-1

4)(A-1)T=(AT)-1

5)/A/A=-1,-1,

3,matrixtranspose

1)(AT)T=A

2)(kA)T=kAT,(kisanyrealnumber)

3)(AB)T=BTAT

4)(A+B)T=AT+BT

4adjointmatrix

1)A*A=AA*=/A/I(AB)*=B*A*

2)(A*)/A/n-2/A**=/=/A/n-1(n=2).

3)(kA)*=knTA*(A*)T=(AT)*

4)ifR(A)=n,thenR(A*)=n

IfR(A)=n-l,thenR(A*)=1

IfR(A)

5)ifAisreversible,(A*)-1=(1//A/A),-1(A*)=(A-l)

*,A*=/A/A-l

5,elementarytransformation(threekinds)

1)changethetwoline(column)

2)withK(k=0)multipliedbyarow(column)inallelements

3)theKofarow(column)ofelementsisdoubledtoanother

line(column)ofthecorrespondingelement

Note:theprimarytransformationisusedtofindtherank,and

therowandcolumntransformationscanbeusedtogether

Inversematrixcanonlybechangedbyroworcolumn

Thesolutionof1inearequationscanonlybechangedbyrow

transformation

6,elementarymatrix

1)amatrixobtainedbyelementarytransformationofaunit

matrix

2)theelementarymatrixPtakestheleft(right)A,andthe

resultingPA(AP)isAandmakesthesamerow(column)

transformationasP

3)elementarymatricesareinvertible,andtheirinverseisof

thesametypeasElementaryMatrices

E-lij=Eij,E(-1),I(k),=Ei(1/k),E(-1),ij(k),=Eij(-k)

Matrixequation7

1)anequationcontaininganunknownmatrix

2)thenecessaryandsufficientconditionsforthesolutionof

matrixequation

AX=Bhas<==>B,eachcolumncanberepresented1inearlybythe

columnvectorofA

<==>r(A)=r(A,B)

Four,questionsandSolutions

1.Propositionsontheconceptandnatureofmatrices

2,theoperationofthematrix(add,multiply,multiply,

transpose)

3,thematrixreversibledecision

NordermatrixA<==>arereversiblematrixofordernB,AB=BA=I

<==>/A/=0

<==>r(A)=n

Thecolumn(row)vectorsofthe<==>Aarelinearlyindependent

<==>Ax=0hasonlyzerosolutions

<==>anyB,Ax=bmakesauniquesolution

Theeigenvaluesof<==>Aarenotzero

4,matrixinversion

1)definethemethod:findB,makeAB=IorBA=I

2:AT=(1/)withthemethodofA*/A)

Note:usingthemethodofinverse,algebraiccofactorinline

typeshouldbeverticallywritteninA*,don'tomitthe

calculationofAij(-1)i+j,whenn>3,usuallywithelementary

transformationmethod.

3)elementarytransformationmethod:(A,I)onlyfor

transformation(I,AT)

4)blockmatrixmethod

5.SolvingmatrixequationAX=B

1)ifAisreversible,thenX=A-1BcanfirstfindA-l,andthen

multiplyA-IBtofindX

2)iftheAisreversible,theprimarytransformationmethod

canbeusedtodirectlyfindtheX

(A,B)(I,Xelementaryrowtransformation)

3)iftheAisnotinvertible,theunknownsequenceequation

canbesetup,andtheGausseliminationmethodisusedasa

ladderequationgroup,andthentheconstantsofeachcolumn

aresolvedrespectively.

Thethirdchapter,linearequations

I.emphasis

1,understanding:vectorvectoroperationsandvectorandthe

linearcombinationoflinearform,theconceptofmaximum

linearlyindependentgroup,theconceptoflineardependence

andlinearindependence,theconceptofvectorgrouprank,

conceptsandpropertiesoftherankofmatrix,theconceptof

solutionsisbased.

2,master:theoperationofthevectorandthelawofoperation,

thecalculationoftherankofthematrix,thestructureofthe

solutionofthehomogeneousandnon-homogeneouslinear

equations.

3,theuseof:1inearcorrelation,linearindependence

judgments,linearequations,thesolutionofthesolution,

homogeneousandnon-homogeneouslinearequationssolution.

Two,difficulties

Decisionoflinearcorrelationandlinearindependence.The

relationbetweentherankofavectorsetandtherankofa

matrix.Therelationbetweenlinearrepresentationandrankof

equationsandvectors.

Three.Analysisofkeyanddifficultpoints

1.Theconceptandoperationofn-dimensionalvectors

1)concept

2)operations

Ifalpha=(Al,A2),...(an)T,beta=(Bl,B2),...BN,T)

Addition:alpha+beta=(al+bl,a2+b2,...(an+bn)T

Numbermultiplication:k=ka2(KAI),…Kan,T)

Internalproduct:(alpha,beta)=albl+a2b2+,…+anbn=Alpha

Tbeta=betaTalpha

2,linearcombinationandlinearlist

3,linearcorrelationandlinearindependence

1)concept

2)thenecessaryandsufficientconditionoflinearcorrelation

andlinearindependence

Linearcorrelation

Alpha1,alpha2,...Linearcorrelationofalphas

<==>homogeneousequations(alpha1,alpha2,...(alphas)(xl,

X2),…(XS)T=0hasnonzerosolutions

Rankr<==>vectorgroup(alpha1,alpha2,...(alpha,s)less

thans(numberofvectors)

Thereisa<==>alphai(i=l,2,...(s)canbelinearlyexpressed

bytherestoftheS-1vectors

Special:nan-dimensionalvectorlinearcorrelation<==>,

alpha1,alpha2...N,alpha=0

N+ln-dimensionalvectorsarelinearlyrelated

Linearlyindependent

Alpha1,alpha2,...Alphasislinearlyindependent

<==>homogeneousequations(alpha1,alpha2,...(alphas)(xl,

X2),…(XS)T=0,onlyzerosolution

Rankr<==>vectorgroup(alpha1,alpha2,...(alphas)=s

(numberofvectors)

<==>everyvectoralphai(i=l,2,...S)cannotbelinearly

representedbytherestoftheS_1vectors

Importantconclusions

TheAandladdervectorsarelinearlyindependent

B,alpha1,alpha2,...Alphasislinearlyindependent,and

anypartofitisalphaII,alphai2,...Thealphaitmustbe

linearlyindependent,andanyofitsextendedgroupsmustbe

linearlyindependent.

ThevectorsofC,22orthogonal,nonzerovectorsmustbe

linearlyindependent.

4,therankofvectorgroupandtherankofmatrix

1)theconceptofmaximallinearlyindependentgroups

2)therankofavectorgroup

3)rankofamatrix

R(A)=R(AT)

TheR(A+B)=R(A)R(B)

TheR(kA)=R(A),k=0

TheR(AB)=min(R(A),R(B))

IfAisreversible,thenR(AB)=R(B);ifBisreversible,

thenR(AB)=R(A)

TheAism*n*Bnarray,Parray,AB=0,R(A)+R(B=n)

4)therelationbetweentherankofavectorsetandtherank

ofamatrix

RowrankofR(A)=A(rankofrowvectorgroupofmatrixA)

=rowrankofA(rankofcolumnvectorgroupofmatrixA)

Therankofthematrixandthevectorgroupareinvariantafter

elementarytransformation

Ifthevectorgroup(I)by(II)lineartable,R(I)=R(II).

Inparticular,theequivalentvectorgroupshavethesamerank,

butthesamevectorsetsarenotnecessarilyequivalent.

5,theconceptandsolutionofthefundamentalsolution

1)concept

2)seekingthelaw

Astheelementaryrowtransformationintotrapezoidalmatrix

ofA,saideachnonzerorowinthefirstnonzerocoefficients

representtheunknownisthemainelement(atotalofR(A)is

amainelement,thentheotherremainingunknown)isafree

variable(atotalofn-R(A)a),thefreevariablesaccording

tostepaftertheassignment,thenyoucangetintothesolving

systemofbasicsolutions.

Thedeterminationofnonzerosolutionsfor6andhomogeneous

equations

1)letAbeam*nmatrix,andAx=0hasanonzerosolution,

andthenecessaryandsufficientconditionisR(A)<n,orA's

columnvectorislinearlyrelated.

2)ifAisnmatrix,Ax=0nonzerosolutionisnecessaryand

sufficientconditionsofA=0,

3)Ax=0,thesufficientconditionofnonzerosolutionism

<n,thatis,thenumberofequationsandthenumberofunknowns

7,thedeterminationofthesolutionsofnonhomogeneouslinear

equations

1)letAbethem*nmatrix,andAx=Bhasthenecessaryand

sufficientconditionthattherankofthecoefficientmatrix

Aisequaltotherankoftheaugmentedmatrix(Aincrease),

thatis,R(A)=R(A+)

2)letAbem*nmatrix,equationAx=b

Thereisauniquesolution<==>R(A)=R(A=n)

Thereareinfinitelymanysolutions<==>R(A)=R(A)

Thesolutionof<==>R(A)+l=r(A)

8,thestructureofsolutionsofnonhomogeneouslinear

equations

SuchasnlinearequationsAx=Bsolution,2,...T,ETAisthe

correspondinghomogeneousequationsAx=0basedsolutions,e

isasolutionofAx=B,KIl+k22+...+ktAx=Bt+zetaETA

isthegeneralsolution.

1)ifE1.2isAx=B,1-,zetazeta2=0Axsolution

2)ifeisAx=B,Ax=0isETAsolutionis+k=BorAxzeta

ETAsolution

3)ifAx=Bhasauniquesolution,thenAx=0hasonlyzero

solutions;conversely,whenAx=0hasonlyzerosolutions,Ax

=Bhasnoinfinitesolution(mayhavenosolutionandmayhave

onlyonesolution)

Four,questionsandSolutions

1.Propositionsabouttheconceptandpropertiesof

n-dimensionalvectors

2.Additionandmultiplicationofvectors

3,linearcorrelationandlinearindependenceproof

1)definitionmethod

LetKIalphal+k2alpha2+...+ks=s=0,thenmakeanidentical

deformationontheupperform(closetotheknowncondition)

B=CcangetAB=AC,soyoucanmultiplyaAontheupperform

bytheinformationoftheknowncondition

Secondly,theupperformisexpandedandtransformeddirectly

intohomogeneouslinearequationsbyknownconditions.Finally,

KIandK2areprovedbyanalysis,...TakethevalueofKSand

drawthedesiredconclusion.

2)usingarank(equaltothenumberofvectors)

3)thehomogeneousequationshaveonlyzerosolutions

4)reductiontoabsurdity

4,therankandthemaximallinearindependentgroupofthe

directionalsetaregiven

Byusingtheprimarytransformationmethod,thevectorgroup

istransformedintoamatrixandsolvedbyelementary

transformation.

5,therankofthematrix

Commonelementarytransformationmethod.

6,solvinghomogeneouslinearequationsandnon-homogeneous

linearequations

Thefourthchapterislinearspace

I.emphasis

1.Understandingtheconceptsoflinearspaces,bases,

dimensions,innerproduct,length,angleanddistance,

orthogonalvectorsandorthonormalbases,orthogonalmatrices

2,master:Rnandtheoperationrulesofthevector.

Calculationofinnerproduct,length,angleanddistance.

3.Use:theorthogonalityoftwovectors.

Two,difficulties

Propertiesandapplicationsoforthogonalmatrices.

Three.Analysisofkeyanddifficultpoints

1.Theconceptsandpropertiesoflinearspacesandbases

2,innerproduct,distanceandincludedangle

1)innerproduct:alpha=beta=albl+a2b2+…+anbn

2)length:"alpha"=(alpha,alpha)=al2+a22+(squareroot...

Thesquarerootof+an2)

3)distance:D="alphabeta”=[(al-bl)2+(A2-B2)2+...+

(an-bn)thesquarerootof2]

4:COS)angletheta=(alpha,beta)/("alpha""beta")

Theta=arccosE(alpha,beta)/("alpha""beta")]

5)orthogonalanglealphaandbetais90degrees,recordedas

analphabeta

Alphaandbetaalpha-beta=0orthogonal<==>

6)orthogonalvectorsets:anytwovectorsareperpendicular

toeachother

Anysetofnonzeroorthogonalvectorsmustbelinearly

independent

ThenumberofvectorsofanynonzeroorthogonalvectorsinRn

isnomorethann

3.Orthogonalizationofvectors

1)theconceptoforthonormalbasis

2)Schmidtorthogonalization(firstorthogonalization,re

integration)

4、orthogonalmatrix

1)concept

2)properties

IftheA/A/==>orthogonalarrayor-1=1

==>A-1isstillanorthogonalarray

==>ifBBT=I,AB=I(AB)T

==>Al=AT

3)thenordermatrixAisthenrowvectoroftheorthogonal

matrix<==>A,whichconstitutesasetofstandardorthogonal

basesofRn

Thencolumnvectorsof<==>Aconstituteasetofstandard

orthogonalbasesofRn

Four,questionsandSolutions

1,determinewhetheragivensetis1inearornot

Generally,itisdeterminedbythedefinitionandpropertyof

linearspace

2,findthebasisanddimensionoflinearspace

3,verifythatthen-dimensionalvectorsetisasetofstandard

orthogonalbasesofRn

Steps:1)thevectorsare22orthogonal,i.e.,theinnerproduct

iszero

2)eachvectorisaunitvector,thatis,thelengthis1

4,calculatetheinnerproductoftwovectors,theangleand

distancebetweenvectors

5,thedirectionofthestandardsetoforthogonal

Steps:1)tojudgethelinearcorrelationofvectors,only

linearlyindependentvectorscanbenormalized

2)orthogonalization(Schmidtorthogonalization)

3)standardVI=1///1//betabeta

6,provethepropositionaboutorthogonalmatrix

7.Thejudgmentoforthogonalmatrix

1)definitionmethod:ifAAT=In,==>Aisorthogonalmatrix

IfAATandInarenotorthogonalarray==>A

Thismethodismostlyusedfortheproofofabstractmatrix.

2)thenordermatrixAisthenrowvector(orcolumnvector)

oftheorthogonalmatrix<==>A,whichconstitutesasetof

standardorthogonalbasesofRn

Therow(column)vectorsof<==>Aareunitvectorsand22

orthogonal

Themethodisusedtogivethematrixofspecificvalues.

Thefifthchapteriseigenvalueandeigenvector

I.emphasis

1.Understanding:theconceptofeigenvaluesandeigenvectors

andtheirbasicproperties.

Theconceptandpropertyofsimilarmatrix,theconditionof

thematrixsimilartothediagonalmatrix.

Jordanmatrix.

2.Master:themethodofcomputingeigenvaluesand

eigenvectors.

Findasimilardiagonalmatrix.

Two,difficulties

Similaritydiagonalizationanditsapplications.

Three.Analysisofkeyanddifficultpoints

1.Conceptsandpropertiesofeigenvaluesandeigenvectorsof

matrices

1)concept

Note:iflambdaistheeigenvaluesofA,thenI-A=0/lambda,

lambda,soI-Aisnotinvertiblematrix

IfafeatureisnotA/I-A/lambdavalueisnotequalto0,

sothelambdaI-Aisaninvertiblematrix

Inparticular,the0istheeigenvaluesofA/A/0<==>A=<==>

irreversible

ThebasicsolutionofAx=0isthe1inearlyindependent

characteristicvectoroflambda=0

FornorderA,ifR(A)=1,thenlambda1=sigmaaii,lambda

2=lambda3=...Lambda=n=0

2)properties

IfXIandX2arecharacteristicvectorscorrespondingtothe

characteristicvaluelambdaI,thenthelinearcombinationof

XI(klxl+k2x2)andX2(nonzero)isstillthecharacteristic

vectoroflambdaI.TheeigenvectorsoflambdaIarenotunique,

andinturn,afeaturevectorcanonlybelongtooneeigenvalue.

Thecharacteristicvectorsofdifferenteigenvaluesare

linearlyindependent,andwhenlambdaIistheKheavy

eigenvaluesofA,AbelongstolambdaI,andthenumberof

linearlyindependenteigenvectorsisnotmorethank.

Thesumofeigenvaluesisequaltothesumofelementsonthe

principaldiagonalofamatrix,andtheproductofthe

eigenvaluesisequaltothevalueofthedeterminantofthe

matrixA.

2,theconceptandpropertyofsimilarmatrix

1)concept

2)properties

IfA~B==>AT~BT

==>Al~B-l(ifAandBarereversible)

==>Ak?Bk(kispositiveinteger)

==>/I-A/lambda=lambda/I-B/AandB,whichhavethesame

characteristicvalue

==>/A/B/,=A,B,andatthesametimereversibleor

irreversible

==>r(A)=R(B)

3.Thenecessaryandsufficientconditionsforthe

diagonalizationofmatrices

1)theconceptofsimilardiagonalization

2)sufficientandnecessarycondition

Aissimilartodiagonalarray.<==>Ahasnlinearlyindependent

eigenvectors

Ineacheigenvalueof<==>A,thenumberoflinearlyindependent

eigenvectorsisexactlyequaltotheeigenvaluesofthe

eigenvalues

3)thesufficientconditionthatAissimilartoadiagonal

matrixisthatAhasndifferenteigenvalues

4,symmetrymatrixsimilarity

1)therealsymmetricmatrixmustbediagonalization

2)characteristics

Theeigenvaluesareal1realnumbers,andthefeaturevectors

arerealvectors

Theeigenvectorsofdifferenteigenvaluesareorthogonalto

eachother

TheKeigenvaluesmusthaveKlinearlyindependenteigenvectors,

ormustber(lambdaI-A)=n-k

Four,questionsandSolutions

1.Themethodoffindingeigenvaluesandeigenvectors

1)pairofabstractmatrices

Basedonthedefinitionandpropertiesofeigenvaluesand

eigenvectors,theeigenvaluevaluesarederived.

2)pairofdigitalmatrices

FromthecharacteristicequationofI-A/lambda=0for

eigenvaluelambdaI(shouldben,containingheavyroot)

Solvingthehomogeneousequationgroup(lambdaI-A)x=0,and

itsbasicsolutionisthelinearlyindependentcharacteristic

vectorcorrespondingtolambda.

2,todeterminewhetherAcanbediagonalization

1)method:nordermatrixAfeaturevectornlinearindependent

diagonalization<==>A

Methodtwo:foranyoftheeigenvaluesofthenordersquare

A,I(setasakiradical)hasN-R(lambdail-A)=ki

2)theprocedureoftransformingAintodiagonalmatrices

First,findtheeigenvaluesofA1,lambda2,...An.

Second,thecorrespondinglinearindependenteigenvectorsxl,

x2,...Xn.

1.

StructureinvertiblematrixP=(xl,X2)...Xn),isP1AP[lambda

=2...]

Lambdan

3.Thedeterminantisobtainedbyeigenvalueandsimilarity

matrix

1)/A/lambda1=lambda2...Lambdan,wherelambda1,lambda

2,...LambdanistheNeigenvalueofA

2)ifA~B/A/B/=,,

4,theuseofsimilardiagonalizationAn

IfthereisaA",invertiblematrixP,theP-1AP=lambda,

then

A=PandAn=PPllambda,lambdaNP1

Whichissimilartoastandardtype:A

5.Theproofofeigenvaluesandeigenvectors

Thesixthchapter,therealtwotimes

I.emphasis

1.Understanding:theconceptofthetwotype,therelation

betweenthetwotypesofsymmetricmatrices,theconceptof

matrixcontract,theconceptofstandardtypeandstandard

standardtype,theconceptofpositivedefinitetypetwoand

theconceptofpositivedefinitematrix.

2:graspthesymmetrymatrixfromthetwoorderandfindthe

twoformfromthesymmetricmatrix.

TherelationshipbetweenthecontractandtakingWestvariable

transformation.

Thejudgementofpositivedefinitetwotypeandpositive

definitematrix.

3,application:orthogonaltransformationmethod,matching

methodandprimarytransformationmethod,twotimesfor

standardtype,fromstandardtypetostandard,standardtype.

Two,difficulties

Thetwotypeisstandardtype.

Three.Analysisofkeyanddifficultpoints

Theconceptsof1andtwotypesandtheirstandardtypes

1)twotimes

Thematrixofthetwotypeisunique,andthequadraticmatrix

shouldbewrittenimmediatelybythetwotype.Onthecontrary,

theChodeMisymmetrymatrixhastobeconstructedtwotimes.

2)thestandardformofthetwotype

Concept

Positiveandnegativeinertialindex,R(f),=r(A)=p+q

Whentheorthogonaltransformationisthestandardtype,the

squarecoefficientofthestandardtypemustbetheNeigenvalue

ofthematrixA,andthemethoddoesnothavethisproperty.

3)inertiatheorem

Thepositiveandnegativeinertialindexofthetwotypeisthe

onlycon