资源描述:
《linearalgebraanditsapplications determinant and trace》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、CHAPTER5DeterminantandTraceInthischapterweshallusetheintuitivepropertiesofvolumetodefinethedeterminantofasquarematrix.Accordingtothepreceptsofelementarygeometry,theconceptofvolumedependsonthenotionsoflengthandangleand,inparticular,perpendicularity,conceptst
2、hatwillbedefinedonlyinChapter8.Nevertheless,itturnsoutthatvolumeisindependentofallthesethings,exceptforanarbitrarymultiplicativeconstantthatcanbefixedbyspecifyingthattheunitcubehavevolumeone.Westartwiththegeometricmotivationandmeaningofdeterminants.Asimplex
3、inll8"isapolyhedronwithn+1vertices.Weshalltakeoneoftheverticestobetheoriginanddenotetherestasa1,...,a,,.Theorderinwhichtheverticesaretakenmatters,sowecall0,ai,....a,,theverticesofanorderedsimplex.Weshallbedealingwithtwogeometricalattributesoforderedsimplice
4、s,theirorientationandvolume.AnorderedsimplexSiscalleddegenerateifitliesonan(n-1)-dimensionalsubspace.Anorderedsimplex(0,at,...,Sthatisnondegeneratecanhaveoneoftwoorientations:positiveornegative.WecallSpositivelyorientedifitcanbedeformedcontinuouslyandnondeg
5、eneratelyintothestandardorderedsimplex(0,et,...,en),whereejisthejthunitvectorinthestandardbasisofR".Bysuchdeformationwemeannvector-valuedcontinuousfunctionsal(t)oft,06、rwiseSiscallednegativelyoriented.ForanondegenerateorientedsimplexSwedefineO(S)as+1or-1,dependingontheorientationofS,andzerowhenSisdegenerate.Thevolumeofasimplexisgivenbytheelementaryformula1Vo1(S)=Vol,,_1(Base)Altitude.(1)nLinearAlgebraandItsApplications,Se
7、condEdition,byPeterD.LaxCopyright((;2007JohnWiley&Sons,Inc.44DETERMINANTANDTRACE45Bybasewemeananyofthe(n-1)-dimensionalfacesofS,andbyaltitudewemeanthedistanceoftheoppositevertexfromthehyperplanethatcontainsthebase.Amoreusefulconceptissignedvolume,denotedasE
8、(S),anddefinedbyE(S)=O(S)Vol(S).(2)SinceSisdescribedbyitsvertices,2(S)isafunctionofa1,...,a,,.Clearly,whentwoverticesareequal,Sisdegenerate,andthereforewehavethefollowing:(i)E(S)=0ifaj=ak,j¢k.Asecondpr