CS545_Lecture_2_matrix algebra refresher

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1、CS545—ContentsIIAdminstrativeDoeseverybodyhavethetextbook?ARefresherofLinearAlgebraVectors,Matrices,NotationInnerProduct(Projections)LinearSystemsEigenvectorsandEigenvaluesTransposes&OuterProductMatrixInversionVectorCalculusReadingAssignmentforNextClassSeeht

2、tp://www-clmc.usc.edu/~cs545AppendixA.LinearAlgebraSincemodellingandcontrolofrobotmanipulatorsrequiresanextensiveuseofmatricesandvectorsaswellasofmatrixandvectoroperations,thegoalofthisappendixistoprovideabrush-upoflinearalgebra.A.IDefinitionsAmatrixofdimensions(mxn)

3、,withmandnpositiveintegers,isanarrayofelementsaijarrangedintomrowsandncolumns:rallaIZaIn]aZIazzaZnA=[aijL=1m=...(A.1)""'..j=1,...,n'.amIamz...amnIfm=n,thematrixissaidtobesquare;ifmnthematrixhasmorerowsthancolumns.Further,ifn=1,t

4、henotation(A.1)isusedtorepresenta(column)vectoraofdimensions(mx1)1;theelementsaiaresaidtobevectorcomponents.AsquarematrixAofdimensions(nxn)issaidtobeuppertriangularifaij=0fori>j:othematrixissaidtobelowertriangularifaij=0fori

5、aij=0fori:/=j,i.e.,o~]=diag{all,azz,...,ann}.oann1Accordingtostandardmathematicalnotation,smallboldfaceisusedtodenotevectorswhilecapitalboldfaceisusedtodenotematrices.Scalarsaredenotedbyromancharacters.336ModellingandControlofRobotManipulatorsIfan(nxn)diagonalmatrixh

6、asallunitelementsonthediagonal(aii=1),thematrixissaidtobeidentityandisdenotedbyIn2•AmatrixissaidtobenullifallitselementsarenullandisdenotedbyO.ThenullcolumnvectorisdenotedbyO.ThetransposeATofamatrixAofdimensions(mxn)isthematrixofdimensions(nxm)whichisobtainedfromtheo

7、riginalmatrixbyinterchangingitsrowsandcolumns:[alla21AT=a.12a22(A.2)alna2nThetransposeofacolumnvectoraistherowvectoraT.An(nxn)squarematrixAissaidtobesymmetricifAT=A,andthusaij=aj;:a1n1a~n.annAn(nxn)squarematrixAissaidtobeskew-symmetricifAT=-A,andthusaij=-ajifori:/=ja

8、ndaii=0,leadingtoa12...a,.1[0...a2nA=-~".-aln-a2n...0Apartitionedmatrixisamatrixwhoseelementsarematrices(blocks)ofproperdimensions: