singular-value-decomposition-fast-track-tutorial.pdf

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1、SingularValueDecomposition(SVD)AFastTrackTutorialDr.EdelGarciaadmin@miislita.comFirstPublishedonSeptember11,2006;LastUpdate:September12,2006CopyrightÓDr.E.Garcia,2006.AllRightsReserved.AbstractThisfasttracktutorialprovidesinstructionsfordecomposingamatri

2、xusingthesingularvaluedecomposition(SVD)algorithm.Thetutorialcoverssingularvalues,rightandlefteigenvectorsandashortcutforcomputingthefullSVDofamatrix.Keywordssingularvaluedecomposition,SVD,singularvalues,eigenvectors,fullSVD,matrixdecompositionProblem:Co

3、mputethefullSVDforthefollowingmatrix:Solution:TTStep1.ComputeitstransposeAandAA.TStep2.DeterminetheeigenvaluesofAAandsorttheseindescendingorder,intheabsolutesense.SquarerootsthesetoobtainthesingularvaluesofA.Step3.ConstructdiagonalmatrixSbyplacingsingula

4、rvaluesindescendingorderalongits-1diagonal.Computeitsinverse,S.TStep4.Usetheorderedeigenvaluesfromstep2andcomputetheeigenvectorsofAA.PlaceTtheseeigenvectorsalongthecolumnsofVandcomputeitstranspose,V.-1TStep5.ComputeUasU=AVS.Tocompletetheproof,computethef

5、ullSVDusingA=USV.TheorthogonalnatureoftheVandUmatricesisevidentbyinspectingtheireigenvectors.Thiscanbedemonstratedbycomputingdotproductsbetweencolumnvectors.Alldotproductsareequaltozero.Alternatively,wecanplottheseandseetheyareallorthogonal.QuestionsFort

6、hematrixT1.ComputetheeigenvaluesofAA.2.ProvethatthisisamatrixofRank2.3.ComputeitsfullSVD.4.ComputeitsRank2Approximation.References1.http://www.miislita.com/information-retrieval-tutorial/svd-lsi-tutorial-3-full-svd.html2.http://www.miislita.com/informati

7、on-retrieval-tutorial/svd-lsi-tutorial-2-computing-singular-values.html3.http://www.miislita.com/information-retrieval-tutorial/svd-lsi-tutorial-1-understanding.htmlCopyrightÓDr.E.Garcia,2006.AllRightsReserved