matrixinversionproblem矩阵的求逆问题

《matrixinversionproblem矩阵的求逆问题》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。

1、Matrixinversionproblem.Givenanon-singularsquarematrix,obtainamatrixsuchthat.Thematrixistheinverseof.Inourcourse,wewouldbeconsideringonlyinversesofnon-singularmatricesoverrealfield(nocomplexmatrix).Matrixinversionthroughitsadjoint.Goodforsmallmatrices.Considerthema

2、trix.Letthecorrespondingcofactormatrixbewhere,…etc.ThentheinverseofisThisschemeisimpracticalforlargematrices.Consequently,weneedeasierapproachestodealwiththeproblem.Wewoulddealwiththesquarematricesonly.Weassumeitsdeterminantisnon-zero.Givenanon-singularsquarematri

3、xwecanobtainitsinverse.Wewillapproachthisproblemfromdifferentangles.a.Usingelementarymatrices:Amatrixisanelementarymatrixifitisobtainedfromanidentitymatrixbyasinglerowoperation.e.gWecangenerateanumberofelementarymatricesfromit.a.b.c.d.Ageneralmatrixcanbeexpresseda

4、sasinglecolumn3rows:LetWhatistheeffectofoperatingbyanelementarymatrixonA?Forinstance,,interchange.isreplacedby,byReplacebyandThesedemonstratetheeffectofelementarymatricesongeneralmatrices--theyeffectivelyachieverow-operations.Therefore,usingsuchmatrices,wecantrans

5、formanon-singularmatrixAintoitsrow-echelonform.Thus,Therefore,thematrixproductThisgivesusaproceduretoobtaininverseofanon-singularmatrixAusingrowtransformation.a.Startwithanaugmentedmatrix.b.Carryoutrow-transformationonthisusingelementarymatrices.c.Whentheleft-side

6、becomesanidentitymatrix,thetransformedrightsidemustbetheinverseoftheoriginalmatrix.Observe:a.Twomatricesandarerowequivalenttoeachotherifonecangetfromusingasequenceofelementarymatricesonthelatter.Thatmeansa.Everyelementaryrow-operationcanbe“undone”byanotherelementa

7、ryrow-operation.Therefore,everyelementarymatrixhasaninverse.b.Theinverseofaproductistheproductoftheinversesinreverseorder.Forinstance,c.Finally,givenanythefollowingstatementsareequivalent:1.hasaninverse.2.hasauniquesolutionforany.3.isrow-equivalentto4.Acanbeexpres

8、sedasproductofelementarymatrices.