矩阵逆学习资料woodbury公式

关于矩阵逆的补充学习资料：Woodburymatrixidentity本文来自维基百科。theWoodburymatrixidentity,namedafterMaxA.Woodbury[1][2]saysthattheinverseofarank-kcorrectionofsomematrixcanbecomputedbydoingarank-kcorrectiontotheinverseoftheoriginalmatrix.Alternativenamesforthisformulaarethematrixinversionlemma,Sherman–Morrison–WoodburyformulaorjustWoodburyformula.However,theidentityappearedinseveralpapersbeforetheWoodburyreport.[3]TheWoodburymatrixidentityis[4]whereA,U,CandValldenotematricesofthecorrectsize.Specifically,Aisn-by-n,Uisn-by-k,Cisk-by-kandVisk-by-n.Thiscanbederivedusingblockwisematrixinversion.InthespecialcasewhereCisthe1-by-1unitmatrix,thisidentityreducestotheSherman–Morrisonformula.InthespecialcasewhenCistheidentitymatrixI,thematrixisknowninnumericallinearalgebraandnumericalpartialdifferentialequationsasthecapacitancematrix.[3]DirectproofJustcheckthattimestheRHSoftheWoodburyidentitygivestheidentitymatrix:DerivationviablockwiseeliminationDerivingtheWoodburymatrixidentityiseasilydonebysolvingthefollowingblockmatrixinversionproblemExpanding,wecanseethattheabovereducestoand,whichisequivalentto.Eliminatingthefirstequation,wefindthat,whichcanbesubstitutedintothesecondtofind.Expandingandrearranging,wehave,or.Finally,wesubstituteintoour,andwehave.Thus,WehavederivedtheWoodburymatrixidentity.DerivationfromLDUdecompositionWestartbythematrixByeliminatingtheentryundertheA(giventhatAisinvertible)wegetLikewise,eliminatingtheentryaboveCgivesNowcombiningtheabovetwo,wegetMovingtotherightsidegiveswhichistheLDUdecompositionoftheblockmatrixintoanuppertriangular,diagonal,andlowertriangularmatrices.NowinvertingbothsidesgivesWecouldequallywellhavedoneittheotherway(providedthatCisinvertible)i.e.Nowagaininvertingbothsides,Nowcomparingelements(1,1)oftheRHSof(1)and(2)abovegivestheWoodburyformulaApplicationsThisidentityisusefulincertainnumericalcomputationswhereA−1hasalreadybeencomputedanditisdesiredtocompute(A+UCV)−1.WiththeinverseofAavailable,itisonlynecessarytofindtheinverseofC−1+VA−1Uinordertoobtaintheresultusingtheright-handsideoftheidentity.IfChasamuchsmallerdimensionthanA,thisismoreefficientthaninvertingA+UCVdirectly.Acommoncaseisfindingtheinverseofalow-rankupdateA+UCVofA(whereUonlyhasafewcolumnsandVonlyafewrows),orfindinganapproximationoftheinverseofthematrixA+Bwherethematrixcanbeapproximatedbyalow-rankmatrixUCV,forexampleusingthesingularvaluedecomposition.Thisisapplied,e.g.,intheKalmanfilterandrecursiveleastsquaresmethods,toreplacetheparametricsolution,requiringinversionofastatevectorsizedmatrix,withaconditionequationsbasedsolution.IncaseoftheKalmanfilterthismatrixhasthedimensionsofthevectorofobservations,i.e.,assmallas1incaseonlyonenewobservationisprocessedatatime.Thissignificantlyspeedsuptheoftenrealtimecalculationsofthefilter.Seealso：Sherman–MorrisonformulaInvertiblematrixSchurcomplementMatrixdeterminantlemma,formulaforarank-kupdatetoadeterminantBinomialinversetheorem;slightlymoregeneralidentity.Notes：1.Jumpup^MaxA.Woodbury,Invertingmodifiedmatrices,MemorandumRept.42,StatisticalResearchGroup,PrincetonUniversity,Princeton,NJ,1950,4ppMR381362.Jumpup^MaxA.Woodbury,TheStabilityofOut-InputMatrices.Chicago,Ill.,1949.5pp.MR325643.^Jumpupto:abHager,WilliamW.(1989).Updatingtheinverseofamatrix.SIAMReview31(2):221–239.doi:10.1137/1031049.JSTOR2030425.MR997457.4.Jumpup^Higham,Nicholas(2002).AccuracyandStabilityofNumericalAlgorithms(2nded.).SIAM.p.258.ISBN978-0-89871-521-7.MR1927606.Press,WH;Teukolsky,SA;Vetterling,WT;Flannery,BP(2007),Section2.7.3.WoodburyFormula,NumericalRecipes:TheArtofScientificComputing(3rded.),NewYork:CambridgeUniversityPress,ISBN978-0-521-88068-8Externallinks：SomematrixidentitiesWeisstein,EricW.,Woodburyformula,MathWorld.imgsrc=//en.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1alt=title=width=1height=1style=border:none;position:absolute;/Retrievedfrom=Woodbury_matrix_identity&oldid=627695139Categories:LinearalgebraLemmasSherman–Morrisonformula（来自维基百科。）StatementSupposeisaninvertiblesquarematrixand,arevectors.Supposefurthermorethat.ThentheSherman–MorrisonformulastatesthatHere,istheouterproductoftwovectorsand.ThegeneralformshownhereistheonepublishedbyBartlett.[5]ApplicationIftheinverseofisalreadyknown,theformulaprovidesanumericallycheapwaytocomputetheinverseofcorrectedbythematrix(dependingonthepointofview,thecorrectionmaybeseenasaperturbationorasarank-1update).Thecomputationisrelativelycheapbecausetheinverseofdoesnothavetobecomputedfromscratch(whichingeneralisexpensive),butcanbecomputedbycorrecting(orperturbing).Usingunitcolumns(columnsfromtheidentitymatrix)foror,individualcolumnsorrowsofmaybemanipulatedandacorrespondinglyupdatedinversecomputedrelativelycheaplyinthisway.[6]Inthegeneralcase,whereisatimesmatrixandandarearbitraryvectorsofdimension,thewholematrixisupdated[5]andthecomputationtakesscalarmultiplications.[7]Ifisaunitcolumn,thecomputationtakesonlyscalarmultiplications.Thesamegoesifisaunitcolumn.Ifbothandareunitcolumns,thecomputationtakesonlyscalarmultiplications.Ver