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1、TheMatrixCookbookKaareBrandtPetersenMichaelSyskindPedersenVersion:January5,2005Whatisthis?Thesepagesareacollectionoffacts(identities,approxima-tions,inequalities,relations,...)aboutmatricesandmattersrelatingtothem.Itiscollectedinthisformfortheconvenienceofanyonewhowantsaquickdes
2、ktopreference.Disclaimer:Theidentities,approximationsandrelationspresentedherewereobviouslynotinventedbutcollected,borrowedandcopiedfromalargeamountofsources.Thesesourcesincludesimilarbutshorternotesfoundontheinternetandappendicesinbooks-seethereferencesforafulllist.Errors:Veryl
3、ikelythereareerrors,typos,andmistakesforwhichweapolo-gizeandwouldbegratefultoreceivecorrectionsatkbp@imm.dtu.dk.Itsongoing:Theprojectofkeepingalargerepositoryofrelationsinvolvingmatricesisnaturallyongoingandtheversionwillbeapparentfromthedateintheheader.Suggestions:Yoursuggestio
4、nforadditionalcontentorelaborationofsometopicsismostwelcomeatkbp@imm.dtu.dk.Acknowledgements:Wewouldliketothankthefollowingfordiscussions,proofreading,extensivecorrectionsandsuggestions:EsbenHoegh-RasmussenandVasileSima.Keywords:Matrixalgebra,matrixrelations,matrixidentities,der
5、ivativeofdeterminant,derivativeofinversematrix,differentiateamatrix.1CONTENTSCONTENTSContents1Basics51.1TraceandDeterminants.......................51.2TheSpecialCase2x2.........................62Derivatives72.1DerivativesofaDeterminant....................72.2DerivativesofanInvers
6、e.......................82.3DerivativesofMatrices,VectorsandScalarForms........92.4DerivativesofTraces.........................112.5DerivativesofStructuredMatrices.................123Inverses143.1ExactRelations............................143.2ImplicationonInverses...............
7、.........143.3Approximations............................153.4GeneralizedInverse..........................153.5PseudoInverse............................154ComplexMatrices174.1ComplexDerivatives.........................175Decompositions205.1EigenvaluesandEigenvectors..............
8、......205.2SingularValueDecomposition..........
