研究生矩阵论资料(双语版) ch2 1-2new

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1、CH2INNERPRODUCTSPACESSEC1--2ByCao,RongmeiEmail:caorongmei@nuaa.edu.cnCellphone:15062258238http://ded.nuaa.edu.cnSEC1INNERPRODUCTSPACES第一节内积空间Sec1InnerproductSpacesWHYDOWENEEDINNERPRODUCT?Innerproductletusbeabletocalculategeometricquantities,suchas,thelengthavector,thean

2、glebetweentwovectors,etc..Theinnerproductletustomeasurehowclosetwovectorsare.Thisisessentialifweneedtostudyconvergence（收敛性），orestimateerror（误差）.3Sec1InnerproductSpacesINNERPRODUCTINANALYTICGEOMETRY?∙?=??cos?=???Withcoordinates,wehave?∙?=?1?1+?2?2+?3?34Sec1Innerprod

3、uctSpacesTHREEKEYELEMENTSOFINNERPRODUCT(1)Positivedefiniteness(2)Symmetry(3)Linearityofscalarprojections(Thescalarprojectionofthesumofthevectorsisthesameasthesumofthescalarprojections;andthescalarprojectionofthemultipleofavectoristhesameasthemultipleofthescalarprojection

4、.)5Sec1InnerproductSpacesDEFOFINNERPRODUCTDef2.1.1LetVbeavectorspaceoverthenumberfieldF(F=RorC).Aninnerproduct(内积)onthevectorspaceVisanoperationthatassignstoeachpairofvectorsuandvinVascalarinFsatisfyingthefollowingconditions:(1)≥0withequalityiffv=0.(Positive-de

5、finiteness)(2)=foralluandvinV.(ConjugatesymmetryorHermitianproperty)(3)=?+?forallu,v,winVandallscalarsa,binF.(Linearityinthefirstargument)6Sec1InnerproductSpacesINNERPRODUCTSPACEAvectorspaceVwithaninnerproductiscalledaninnerproductspace（内积空间

6、）.AninnerproductspaceovertherealnumberfieldRiscalledaEuclideanspace,andaninnerproductspaceoverthecomplexnumberfieldCiscalledaunitaryspace(酉空间).7Sec1InnerproductSpacesLENGTHOFAVECTOR(向量长度)Withinnerproduct,wecandefinelengthsofvectors.Ifuandvarevectorsinaninnerproductspace

7、V,Thelengthornorm（范数）ofvisgivenby?=If?=1,thenviscalledaunitvector（单位向量）.8Sec1InnerproductSpacesSCALARPROJECTION（数量投影）Withinnerproduct,wecandefinescalarprojection.IfuandvarevectorsinaninnerproductspaceVand?≠?,thenthescalarprojectionofuontovisgivenby?uv9Sec1Inner

8、productSpacesVECTORPROJECTION（向量投影）Withinnerproduct,wecandefinevectorprojection.Ifuandvarevectorsinanin