MIT18_06SCF11_Ses1.2An Overview of Key Ideas

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1、EliminationwithmatricesMethodofEliminationEliminationisthetechniquemostcommonlyusedbycomputersoftwaretosolvesystemsoflinearequations.ItfindsasolutionxtoAx=bwheneverthematrixAisinvertible.Intheexampleusedinclass,⎡⎤⎡⎤1212A=⎣381⎦andb=⎣12⎦.0412Thenumber1intheupperleftcornerofAiscall

2、edthefirstpivot.Werecopythefirstrow,thenmultiplythenumbersinitbyanappropriatevalue(inthiscase3)andsubtractthosevaluesfromthenumbersinthesecondrow.Thefirstnumberinthesecondrowbecomes0.Wehavethuseliminatedthe3inrow2column1.Thenextstepistoperformanothereliminationtogeta0inrow3column1

3、;herethisisalreadythecase.Thesecondpivotisthevalue2whichnowappearsinrow2column2.Wefindamultiplier(inthiscase2)bywhichwemultiplythesecondrowtoeliminatethe4inrow3column2.Thethirdpivotisthenthe5nowinrow3column3.WestartedwithaninvertiblematrixAandendedwithanuppertriangularmatrixU;th

4、elowerleftportionofUisfilledwithzeros.Pivots1,2,5areonthediagonalofU.⎡⎤⎡⎤⎡⎤121121121A=⎣381⎦−→⎣02−2⎦−→U=⎣02−2⎦041041005��2Werepeatthemultiplicationsandsubtractionswiththevectorb=12.2Forexample,wemultiplythe2inthefirstpositionby3andsubtractfrom12toget6inthesecondposition.Whencalcul

5、atingbyhandwecandothisefficientlybyaugmentingthematrixA,appendingthevectorbasafourthorfinalcolumn.ThemethodofeliminationtransformstheequationAx=binto��121anewequationUx=c.Intheexampleabove,U=02−2comesfrom005��2Aandc=6comesfromb.−10TheequationUx=ciseasytosolvebybacksubstitution;in

6、ourexample,z=−2,y=1andx=2.ThisisalsoasolutiontotheoriginalsystemAx=b.1ThedeterminantofUistheproductofthepivots.Wewillseethisagain.Pivotsmaynotbe0.Ifthereisazerointhepivotposition,wemustexchangethatrowwithonebelowtogetanon-zerovalueinthepivotposition.Ifthereisazerointhepivotposi

7、tionandnonon-zerovaluebelowit,thenthematrixAisnotinvertible.Eliminationcannotbeusedtofindauniquesolutiontothesystemofequations–itdoesn’texist.EliminationMatricesTheproductofamatrix(3x3)andacolumnvector(3x1)isacolumnvector(3x1)thatisalinearcombinationofthecolumnsofthematrix.Thepr

8、oductofarow(1x3)andamatrix(3x3)isarow(1x3)thatisalinea