矩阵的初等变换及其应用Elementary transformation of matrix and its application.docx
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矩阵的初等变换及其应用Elementary transformation of matrix and its application.docx
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矩阵的初等变换及其应用Elementarytransformationofmatrixanditsapplication
矩阵的初等变换及其应用(Elementarytransformationofmatrixanditsapplication)
Elementarytransformationofmatrixanditsapplication
WangDan
Elementarytransformationofmatrixanditsapplication
Abstract
Elementarytransformationofmatrixisanimportantmethodofstudyingmatrix,anditisthecoreofapplicationinlinearalgebra.Thispaperintroducessomeconceptsandpropertiesassociatedwiththematrix,onthebasisofmatrixrank,thebasisforjudgmentmatrixisinvertible,afterinversematrixequations,eigenvaluesandeigenvectors,twotypesofstandardform,andillustratetheapplicationofelementarytransformationofmatrixintheaboveishowtoplaytheroleof.
Keywords:
matrix,elementarytransformation,application
The,elementary,transformation,of,matrix,and,its,applications
Abstract
ElementarytransformationmatrixisanimportantmeansofMatrixisthecorelinearalgebraapplications.Thisarticlebrieflydescribessomeoftheconceptsandpropertiesassociatedwiththematrixasabasis,therankofamatrixtodeterminewhetheramatrixisreversibleafterinversematrix,seekingbasicsolutionslineequationsfindeigenvalues,andeigenvectors,quadraticstandardShapeandsoon.Illustratetheelementarytransformationmatrixintheaboveapplicationsishowtoplayarole.
Keywords:
matrix,elementary,transformation,application
Catalog
1.introduction6
2.therelatedconceptsofmatrix7
2.1definitionofmatrix7
2.2transposeofmatrix7
2.3elementarytransformationofmatrixandelementarymatrix7
3.theapplicationofelementarytransformationofmatrix8
3.1,therankofthematrix8
3.2theinversematrixofthematrix10
3.3usingelementarytransformationtosolvematrixequation11
3.4findthesolutionoflinearequations12
Theconditionsfortheexistenceofnonzerosolutionsof3.4.1homogeneouslinearequationsare13
Conditionsfortheexistenceofsolutionsof3.4.2nonhomogeneouslinearequations14
3.5findtheeigenvaluesandeigenvectorsofthematrix15
3.6,useelementarytransformation,twotimesasstandardtype17
Summary19
References19
1.introduction
Inthecourseofstudyinglinearalgebra,Ifindthattheelementarytransformationofmatrixisveryextensiveandrunsthroughthewholechapter.Itisthekeytosolvetheprobleminlinearalgebra.Linearequationsisthebeginningoftheelementarytransformationmatrix,thematrixeffectcanalsobesaidtobeoflinearalgebra,eachknowledgepointoflinearalgebraandlinearalgebraandmatrixarecloselyrelated,eachinmathematicsbothcanplayarole.Biology,economics,physics,cryptographyrequiresknowledgeofmathematics,thesignificanceofmatrixelementarytransformationofmatrix,ascanbeimagined,isthecomplexmatrixintoasimpleformiseasytocalculateandunderstand.
Inreallife,manyaspectsinvolvetheknowledgeofmatrices,
Instudyingthevirtualaircraftmodel,wewillfindthattheoperationofthematrixplaysacrucialrole.Theplanesurfaceappearstobesmooth,butthegeometricstructureisperplexing,theflowequationismoredifficult,mustalsoconsiderotherexternalfactors,butweusethematrixknowledgetobeabletosolvetheproblemverywell.Therearemanyotherapplications,forexample,matrixeigenvaluesandeigenvectorsisthekeytosolvemanyproblemsinphysics,mechanicsandengineeringtechnology;nowthegamecompanyandBankAccountconfidentialsecurity,butalsotheuseofmatrixtheoryinventedthematrixcard;simulationinequipmentmonitoringsysteminengineering,radioandtelevision;largescreendisplayworks,TVteaching,commandandcontrolcenteretc.mainlyusedmatrixswitcherandsoon.
2.conceptsrelatedtomatrices
2.1definitionofmatrix
Tableisarectangularmatrix.Similartothecrossandthedeterminantiscalledarow,calledverticalcolumns,withaline,thelineandthelineandtherowelementmatrixforshortnote.
Transposeofthe2.2matrix
Letamatrixbecalledamatrix
Forthetransposeofthematrix,remember
Elementarytransformationandelementarymatrixof2.3matrices
1,thefollowingthreetransformationscalledmatrices,calledthematrixoftheprimaryrow(column)transform,collectivelyreferredtoastheelementarytransformationofthematrix:
(1)thetworow(column)oftheexchangematrix
(2)theelementsofarow(column)ofamatrixaremultipliedbyanonzeroconstant
(3)aconstantoftheelementsofarow(column)ofamatrixaddedtothecorrespondingelementofanotherrow(column)
Elementaryrowandcolumntransformationsarecollectivelyreferredtoaselementarytransformations
2.Thematrixobtainedbyelementarytransformationofaunitmatrixiscalledelementarymatrix.
Threetypesofelementarymatrices:
(1)elementarycommutativematrices:
thesecondandthesecondlinesofthecommutativeunitmatrix
(2)theelementarymultipliedmatrix:
therow(column)oftheunitmatrixtakesthenonzeroconstant,i.e.
(3)elementarydoublymatrix:
thefirstrowofaunitmatrixisaddedtothefirstline,orthefirstrowismultipliedtothenextcolumn
Ifthematrixistransformedintoamatrixbyafiniteelementarytransformation,itissaidtobeequivalent
3,matrixequivalencehasthefollowingproperties:
(1)reflexivity,thatis,theselfequivalenceofanymatrix;
(2)symmetry,thatis,theequivalenceofanymatrix,ifandequivalence;
(3)transitivityisequivalenttoanymatrix,andifandequivalence,equivalence,andequivalence;
Theapplicationofelementarytransformationof3.matrices
3.1,therankofthematrix
Manymethodsformatrixrank,generaldefinitionmethod,elementarytransformationmethod,formulamethodandcomprehensivemethod,butwhenthespecificelementofthematrixisknown,usingelementarytransformationmethodisfornonzerorow(column)number.
Thehighestorderofanonzerodivisordefinedina3.1.1matrixiscalledtherankofamatrix.Thatis,thereisarankorderofno0,andallordersofvariables(ifany)are0,thentherankofthematrixis(or/orrank)
(1)
(2)therankofthezeromatrixis0
(3)therankofaladdermatrix=thenumberofnonzerorowsinarow
Theorem3.1.1theelementarytransformationofamatrixdoesnotchangetherankofamatrix
Theorem3.1.2rowrankofamatrix=rowrankofamatrix
Theorem3.1.
3,theequivalentmatriceshavethesamerank,buttheirinverseisnottrue,thatis,thematriceswiththesamerankmaynotbeequivalent,andthematricesofthesametypeandthesamerankareequivalenttoeachother
Findtherankofamatrix,andgiveabriefintroductionofthemostcommonmethod:
(1)definitionmethod:
Ifthematrixhasanonzeroorder,andallthesuborders(ifany)areall0,then.
Ifthereisanonzeroorderinthematrix,andalloftheordervariablescontainingthisorderare0.
Theusageofmatrixrankcanbecalculatedwithsimpleformulaomitalot.
(2)thenumberofzerorowsinCentralAfricaistherankofthematrix.
Thisisbecausetheelementaryrowtransformationdoesnotchangetherankofthematrix,inaddition,itcanbetransformedintoacolumnladderrankbytheelementarycolumntransformation,andtheelementarytransformationcanbeusedasthestandardformtoobtaintherank.
Example1findtherankofamatrix.
Solution1:
takethe2orderoftheupperleftofthematrix
However,thereareonly3linesinthematrix,soitisnecessarytofindthe3orderofthevariablescontainedinthematrix.
Solution2:
todoelementaryrowtransformation
Duetonon-zerobehavior2.
Itcanbeseenthatthedefinitionmethodisonlysuitableforthecalculationofsimplematrix,butifitisahigherordermatrix,itisveryinconvenienttocalculate.
3.2,theinversematrixofthematrix
Thedefinitionof3.2.1issetasasquarematrix,iftheordermatrixexists
Hereistherankunitmatrix,whichiscalledtheinvertiblematrix,andiscalledtheinversematrix.
Note
(1)ifitisinvertible,itsinversematrixisunique,andtheinversematrixis;
(2)theinvertibleproblemofthematrixisthecaseoftheopponent'smatrix.
Settheinvertiblematrixoforder,andtheinversematrixisasfollows:
Example2issetupasainvertiblesquarematrix,andtheresultingmatrixisdenotedbythefollowinglineandcolumn
(1)provedtobereversible;
(2)seeking
Proof:
(1)sincetheleftmultiplicationoftheelementarymatrixcorrespondstothetworowsoftheinterchange,sothereis
Because,sothematrixisreversible
(2)
Example3usestheelementarytransformationofthematrixtofindtheinversematrixofthematrix
Solution:
so
Inshort,weintheinversematrixwithelementarytransformation,wemustfirstselectedbyelementaryrowtransformationorelementarycolumntransformation,notethatifusingelementaryrowtransformationmustbefromfirsttolastbyelementaryrowtransformation,usingelementarycolumntransformationmustbefromfirsttolastbyelementarycolumntransformation.
Butintheinversedoesnotneedtocheckwhetherthereversiblematrix,elementarytransformationcanbedirectlyobtained,ifthesimplestformofasquarematrixtransformunitisnotleftaftertheshow,theoriginalmatrixisirreversible.
3.3usingelementarytransformationtosolvematrixequation
(1)ifitisreversible,then
(2)ifitisreversible,then
(3)ifbotharereversible,then
Firstofall
Again
Thiscanbeobtained
Thematrixequationsoftypecanonlybeelementaryrowtransformations(ontheleft);thepaircanonlybeelementarycolumntransformations(ontheright)
Example4solvingmatrixequation
Solution:
lettheoriginalequationbe...
therefore
3.4solvingthesys
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