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1、Chapter6JordancanonicalformofmatricesInthepreviouschapterwehavediscussedtheproblemofreducingmatrixestodiagonalmatrices.Ingeneral,matricesthatcannotbereducedtodiagonalmatricescanbereducedtoblockdiagonalmatrices,ortheJordancanonicalform.Inthischapterweshallconsiderthispro
2、blem,specificallythefollowing:1.thenecessaryandsufficientconditionfor-matricestobeequivalent.2.methodsoffindingJordancanonicalforms.Thischapterisdividedintofoursections.Thesecondandthirdsectionsdealwiththefirstproblem,theothersectionswiththesecondproblem.6.1.Necessaryan
3、dsufficientconditionfortwomatricestobesimilarForagivenmatrix,howdoesonefindasimplematrixsuchthat~?Weproceedasfollows.Firstfindthenecessaryandsufficientconditionfortwomatricestobesimilar,andthenusingthesufficientconditionfindthematrixrequired.Supposetwomatricesandaresimi
4、lar,i.e.,~.Thenthereexistsmatrixsuchthat.Thus.Weeasilyproveisequivalentto,i.e.,.Conversely,if,wewillprovethatthereexistconstantmatricesandsuchthat.Consequently,,andhence~.Thuswehave:Theorem1.twomatricesandaresimilarifandonlyiftheircharacteristicmatricesareequivalent:.Th
5、istheoremisveryimportant.Itnotonlygivesthenecessaryandsufficientconditionfortwomatricestobesimilar,but,moreimportantly,itchangesthesimilarityrelationintoanequivalencerelation.Itisdifficultforustodealwithasimilarityrelation,but,usingelementaryoperations,wecanmaketheequiv
6、alencerelationmorespecific.Howeverwehavegivenonlytheconclusionofthetheorem,noitsproof.Herethecharacteristicmatricesare-matrices.However,theequivalenceconceptof-matriceshasnotbeengiveneither.Thereforebeforeaproofoftheabovetheoremwefirsthavetodefinesomeconcepts,suchasthec
7、onceptsofelementaryoperationson-matrices,theconceptofequivalenceoftwo-matrices.Theconceptoftherankofa-matricesisthesameasthatofaconstantmatrix.Thus,thehighestorderoftheminorsofnotbeingidenticallyzeroiscalledtherankofa-matrices.Itistobenotedthatminorsofarepolynomialsin.B
8、yapolynomialinbeingidenticallyzerowemeanthatanynumbercanbeitsroot.Henceitscoefficientsareallzero,andweoftensay
