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1、TheMatrixCookbook[http://matrixcookbook.com]KaareBrandtPetersenMichaelSyskindPedersenVersion:November14,2008Whatisthis?Thesepagesareacollectionoffacts(identities,approxima-tions,inequalities,relations,...)aboutmatricesandmattersrelatingtothem.Itiscoll
2、ectedinthisformfortheconvenienceofanyonewhowantsaquickdesktopreference.Disclaimer:Theidentities,approximationsandrelationspresentedherewereobviouslynotinventedbutcollected,borrowedandcopiedfromalargeamountofsources.Thesesourcesincludesimilarbutshorter
3、notesfoundontheinternetandappendicesinbooks-seethereferencesforafulllist.Errors:Verylikelythereareerrors,typos,andmistakesforwhichweapolo-gizeandwouldbegratefultoreceivecorrectionsatcookbook@2302.dk.Itsongoing:Theprojectofkeepingalargerepositoryofrela
4、tionsinvolvingmatricesisnaturallyongoingandtheversionwillbeapparentfromthedateintheheader.Suggestions:Yoursuggestionforadditionalcontentorelaborationofsometopicsismostwelcomeatcookbook@2302.dk.Keywords:Matrixalgebra,matrixrelations,matrixidentities,de
5、rivativeofdeterminant,derivativeofinversematrix,differentiateamatrix.Acknowledgements:Wewouldliketothankthefollowingforcontributionsandsuggestions:BillBaxter,BrianTempleton,ChristianRishøj,ChristianSchr¨oppelDouglasL.Theobald,EsbenHoegh-Rasmussen,Glynn
6、eCasteel,JanLarsen,JunBinGao,J¨urgenStruckmeier,KamilDedecius,KorbinianStrim-mer,LarsChristiansen,LarsKaiHansen,LelandWilkinson,LiguoHe,LoicThibaut,MiguelBar˜ao,OleWinther,PavelSakov,StephanHattinger,VasileSima,VincentRabaud,ZhaoshuiHe.Wewouldalsolike
7、thankTheOticonFoundationforfundingourPhDstudies.1CONTENTSCONTENTSContents1Basics51.1TraceandDeterminants.......................51.2TheSpecialCase2x2.........................52Derivatives72.1DerivativesofaDeterminant....................72.2Derivativeso
8、fanInverse.......................82.3DerivativesofEigenvalues......................92.4DerivativesofMatrices,VectorsandScalarForms........92.5DerivativesofTraces.........................112.6Derivativesofvectornorms.....................132.7De
