资源描述：
《elements of multilinear algebra》由会员上传分享，免费在线阅读，更多相关内容在工程资料-天天文库。
1、September21,20158:8Matrices:Algebra,AnalysisandApplications-9inx6inb2108-ch05page325Chapter5ElementsofMultilinearAlgebra5.1TensorProductofTwoFreeModulesLetDbeadomain.RecallthatNiscalledafreefinitedimensionalmoduleoverDifNhasafinitebasise1,...,en,i.e.dimN=n.ThenN:=Hom(N,D)isafreen-di
2、mensionalmodule.FurthermorewecanidentifyHom(N,D)withN.(SeeProblem1.)Definition5.1.1LetM,Nbetwofreefinitedimensionalmod-ulesoveranintegraldomainD.ThenthetensorproductM⊗NisidentifiedwithHom(N,M).Moreover,foreachm∈M,n∈Nweidentifym⊗n∈M⊗DNwiththelineartransformationm⊗n:N→Mgivenbyf→f(n)
3、mforanyf∈N.Proposition5.1.2LetM,NbefreemodulesoveradomainDwithbases[d1,...,dm],[e1,...,en]respectively.ThenM⊗DNisafreemodulewiththebasisdi⊗ej,i=1,...,m,j=1,...,n.InparticulardimM⊗N=dimMdimN.(5.1.1)(SeeProblem3.)ForanabstractdefinitionofM⊗DNforanytwoD-modulesseeProblem16.325Septembe
4、r21,20158:8Matrices:Algebra,AnalysisandApplications-9inx6inb2108-ch05page326326MatricesIntuitively,oneviewsM⊗Nasalinearspanofallelementsoftheformm⊗n,wherem∈M,n∈Nsatisfythefollowingnaturalproperties:1.a(m⊗n)=(am)⊗n=m⊗(an)foralla∈D.2.(a1m1+a2m2)⊗n=a1(m1⊗n)+a2(m2⊗n)foralla1,a2∈D.(Line
5、arityinthefirstvariable.)3.m⊗(a1n1+a2n2)=a1(m⊗n1)+a2(u⊗n2)foralla1,a2∈D.(Linearityinthesecondvariable.)Theelementm⊗niscalledadecomposabletensor,ordecomposableelement(vector),orrankonetensor.Proposition5.1.3LetM,NbefreemodulesoveradomainDwithbases[d1,...,dm],[e1,...,en]respectively.T
6、henanyτ∈M⊗DNisgivenbyi=m,j=nτ=ad⊗e,A=[a]∈Dm×n.(5.1.2)ijijiji=j=1Let[u1,...,um],[v1,...,vn]bedifferentbasesofM,Nrespectively.m,nm×nAssumethatτ=i,j=1bijui⊗vjandletB=[bij]∈D.ThenB=PAQT,wherePandQarethetransitionmatricesfromthebases[d1,...,dm]to[u1,...um]and[e1,...,en]to[v1,...,vn].(T
7、hatis,[d1,...,dm]=[u1,...um]P,[e1,...,en]=[v1,...,vn]Q.)SeeProblem6.Definition5.1.4LetM,NbefreefinitedimensionalmodulesoveradomainD.Letτ∈M⊗DNbegivenby(5.1.2).Therankofτ,denotedbyrankτ,istherankoftherepresentationmatrixA,i.e.rankτ=rankA.Thetensorrankofτ,denotedbyRankτ,isktheminimalks
8、uchthatτ=l=1ml⊗nlforsomeml
