linearalgebraanditsapplications determinant and trace

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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,0

6、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