矩阵分析课件 Review of basic concepts and facts in linear.ppt

Reviewofbasicconceptsandfactsinlinearalgebra

guidelinePrerequisite:Text:MatrixAnalysis,RogerA.Horn,CharlesR.Johnson,CambridgeUniversityPress,Reprintedition(February23,1990).ISBN0521386322,575pHomeworkassignment:weeklyonFridayExam:finalexam(openbook)onOct.,2014(temporary)Content:vectorspace,norms,eigenvalues,unitarymatrix,Hermitianmatrix,matrixfactorization,canonicalform,nonnegativematrixQQ群：314073497

Definition:VectorSpace(V,?+;F)Avectorspace(overafieldF)consistsofasetValongwith2operations‘+’and‘’s.t.Forthevectoraddition+: ?v,w,u?Vv+w?V (Closure) v+w=w+v (Commutativity)(v+w)+u=v+(w+u) (Associativity)?0?Vs.t.v+0=v (Zeroelement)??v?Vs.t.v?v=0 (Inverse)(2)Forthescalarmultiplication: ?v,w?Vanda,b?F, av?V (Closure)(a+b)v=av+bv (Distributivity)a(v+w)=av+aw(a?b)v=a(bv)=abv (Associativity)1v=v(Scalaridentityofmultiplication)

Expressionofforce,velocity,gradient,

SubspaceAsetUisasubspaceofavectorspaceVifEveryelementofUisinV,andUisavectorspace.

LinearCombinationsConsiderasetofvectors{v1,….,vn}andasetofscalars{a1,…,an}Alinearcombinationofthevectorsisa1v1+a2v2+…+anvnRemark:Vectorspace=Collectionoflinearcombinationsofvectors.

Definition:SpanLetS={s1,…,sn|sk?(V,+,R)}beasetofnvectorsinvectorspaceV.ThespanofSisthesetofalllinearcombinationsofthevectorsinS,i.e.,withLemma:Thespanofanysubsetofavectorspaceisasubspace.Proof:LetS={s1,…,sn|sk?(V,+,R)}and?QEDNote:spanSisthesmallestvectorspacecontainingallmembersofS.

Example:Proof:Theproblemistantamounttoshowingthatforallx,y?R,?uniquea,b?Rs.t.i.e.,hasauniquesoluti