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1、MatrixDi erentiation(andsomeotherstu )RandalJ.BarnesDepartmentofCivilEngineering,UniversityofMinnesotaMinneapolis,Minnesota,USA1IntroductionThroughoutthispresentationIhavechosentouseasymbolicmatrixnotation.Thischoicewasnotmadelightly.Iamastrongadvocateofindexnotation,whenappropriate.Fore
2、xample,indexnotationgreatlysimpli esthepresentationandmanipulationofdi erentialgeometry.Asarule-of-thumb,ifyourworkisgoingtoprimarilyinvolvedi erentiationwithrespecttothespatialcoordinates,thenindexnotationisalmostsurelytheappropriatechoice.Inthepresentcase,however,Iwillbemanipulatinglar
3、gesystemsofequationsinwhichthematrixcalculusisrelativelysimplywhilethematrixalgebraandmatrixarithmeticismessyandmoreinvolved.Thus,Ihavechosentousesymbolicnotation.2NotationandNomenclatureDe nition1Letaij2R,i=1,2,...,m,j=1,2,...,n.Thentheorderedrectangulararray23a11a12a1n676a21a22a2
4、n7A=6......7(1)674...5am1am2amnissaidtobearealmatrixofdimensionmn.WhenwritingamatrixIwilloccasionallywritedownitstypicalelementaswellasitsdimension.Thus,A=[aij],i=1,2,...,m;j=1,2,...,n,(2)denotesamatrixwithmrowsandncolumns,whosetypicalelementisaij.Note,the rstsubscriptlocatestherowin
5、whichthetypicalelementlieswhilethesecondsubscriptlocatesthecolumn.Forexample,ajkdenotestheelementlyinginthejthrowandkthcolumnofthematrixA.De nition2Avectorisamatrixwithonlyonecolumn.Thus,allvectorsareinherentlycolumnvectors.Convention1Multi-columnmatricesaredenotedbyboldfaceuppercaselett
6、ers:forexample,A,B,X.Vectors(single-columnmatrices)aredenotedbyboldfacedlowercaseletters:forexample,a,b,x.Iwillattempttouselettersfromthebeginningofthealphabettodesignateknownmatrices,andlettersfromtheendofthealphabetforunknownorvariablematrices.1CE8361Spring2006Convention2Whenitisuseful
7、toexplicitlyattachthematrixdimensionstothesymbolicnotation,Iwilluseanunderscript.Forexample,A,indicatesaknown,multi-columnmatrixwithmrowsmnandncolumns.AsuperscriptTdenotesthematrixtransposeoperation;forexample,ATdenotesthetransposeofA.Similarly,ifAhasaninverseitwillbeden
