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1、MatrixDi erentiation(andsomeotherstu )RandalJ.BarnesDepartmentofCivilEngineering,UniversityofMinnesotaMinneapolis,Minnesota,USA1IntroductionThroughoutthispresentationIhavechosentouseasymbolicmatrixnotation.Thischoicewasnotmadelightly.Iamastrongadvocate
2、ofindexnotation,whenappropriate.Forexample,indexnotationgreatlysimpli esthepresentationandmanipulationofdi erentialgeometry.Asarule-of-thumb,ifyourworkisgoingtoprimarilyinvolvedi erentiationwithrespecttothespatialcoordinates,thenindexnotationisalmostsu
3、relytheappropriatechoice.Inthepresentcase,however,Iwillbemanipulatinglargesystemsofequationsinwhichthematrixcalculusisrelativelysimplywhilethematrixalgebraandmatrixarithmeticismessyandmoreinvolved.Thus,Ihavechosentousesymbolicnotation.2NotationandNomen
4、clatureDe nition1Letaij2R,i=1,2,...,m,j=1,2,...,n.Thentheorderedrectangulararray23a11a12a1n676a21a22a2n7A=6......7(1)674...5am1am2amnissaidtobearealmatrixofdimensionmn.WhenwritingamatrixIwilloccasionallywritedownitstypicalelementaswellasitsdi
5、mension.Thus,A=[aij],i=1,2,...,m;j=1,2,...,n,(2)denotesamatrixwithmrowsandncolumns,whosetypicalelementisaij.Note,the rstsubscriptlocatestherowinwhichthetypicalelementlieswhilethesecondsubscriptlocatesthecolumn.Forexample,ajkdenotestheelementlyinginthej
6、throwandkthcolumnofthematrixA.De nition2Avectorisamatrixwithonlyonecolumn.Thus,allvectorsareinherentlycolumnvectors.Convention1Multi-columnmatricesaredenotedbyboldfaceuppercaseletters:forexample,A,B,X.Vectors(single-columnmatrices)aredenotedbyboldfaced
7、lowercaseletters:forexample,a,b,x.Iwillattempttouselettersfromthebeginningofthealphabettodesignateknownmatrices,andlettersfromtheendofthealphabetforunknownorvariablematrices.1CE8361Spring2006Convention2Whenitisusefultoexplicitlyattachthematrixdimension
8、stothesymbolicnotation,Iwilluseanunderscript.Forexample,A,indicatesaknown,multi-columnmatrixwithmrowsmnandncolumns.AsuperscriptTdenotesthematrixtransposeoperation;forexample,ATdenotesthetransposeofA.Similarly,ifAhasaninverseitwillbeden
