Matrix Exponential -- from Wolfram MathWorld

Matrix exponentials are important in the solution of systems of ordinary differential equations (e.g., Bellman 1970). In some cases, it is a simple matter to ... Algebra AppliedMathematics CalculusandAnalysis DiscreteMathematics FoundationsofMathematics Geometry HistoryandTerminology NumberTheory ProbabilityandStatistics RecreationalMathematics Topology AlphabeticalIndex NewinMathWorld Algebra LinearAlgebra Matrices MatrixOperations MathWorldContributors Rowland,Todd MatrixExponential Download Wolfram Notebook Thepowerseriesthatdefinestheexponentialmapalsodefinesamapbetweenmatrices. Inparticular, (1) (2) (3) convergesforanysquarematrix,whereistheidentity matrix.ThematrixexponentialisimplementedintheWolfram LanguageasMatrixExp[m]. TheKroneckersumsatisfiestheniceproperty (4) (HornandJohnson1994,p. 208). Matrixexponentialsareimportantinthesolutionofsystemsofordinarydifferentialequations(e.g.,Bellman1970). Insomecases,itisasimplemattertoexpressthematrixexponential.Forexample,whenisadiagonal matrix,exponentiationcanbeperformedsimplybyexponentiatingeachofthe diagonalelements.Forexample,givenadiagonalmatrix (5) Thematrixexponentialisgivenby (6) Sincemostmatricesarediagonalizable, itiseasiesttodiagonalizethematrixbeforeexponentiatingit. Whenisanilpotent matrix,theexponentialisgivenbyamatrix polynomialbecausesomepowerofvanishes.For example,when (7) then (8) and. Forthezeromatrix, (9) i.e.,theidentitymatrix.Ingeneral, (10) sotheexponentialofamatrixisalwaysinvertible,withinversetheexponentialofthenegativeofthematrix.However,ingeneral,theformula (11) holdsonlywhenandcommute, i.e., (12) Forexample, (13) while (14) Evenforageneralreal matrix,however,thematrixexponentialcanbequitecomplicated (15) where (16) (17) (18) (19) and (20) As,thisbecomes (21) SeealsoExponentialFunction,ExponentialMap,Kronecker Sum,Matrix,MatrixPower PortionsofthisentrycontributedbyTodd Rowland ExplorewithWolfram|Alpha Morethingstotry: matrixoperations conjugatetranspose matrixexponential[{ln(1/2),0},{0,1}] ReferencesBellman,R. E.IntroductiontoMatrixAnalysis,2nded.NewYork:McGraw-Hill,1970.Horn, R. A.andJohnson,C. R.Topics inMatrixAnalysis.Cambridge,England:CambridgeUniversityPress,p. 208, 1994.Moler,C.andvanLoan,C."NineteenDubiousWaystoCompute theExponentialofaMatrix,Twenty-FiveYearsLater."SIAMRev.45, 3-49,2003.ReferencedonWolfram|AlphaMatrixExponential Citethisas: Rowland,ToddandWeisstein,EricW."MatrixExponential."FromMathWorld--A WolframWebResource.https://mathworld.wolfram.com/MatrixExponential.html Subjectclassifications Algebra LinearAlgebra Matrices MatrixOperations MathWorldContributors Rowland,Todd Created,developedandnurturedbyEricWeissteinatWolframResearch