The Matrix Cookbook.pdf
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TheMatrixCookbook
[http://matrixcookbook.com]
KaareBrandtPetersen
MichaelSyskindPedersen
Version:November14,2008
Whatisthis?Thesepagesareacollectionoffacts(identities,approxima-
tions,inequalities,relations,...)aboutmatricesandmattersrelatingtothem.
Itiscollectedinthisformfortheconvenienceofanyonewhowantsaquick
desktopreference.
Disclaimer:Theidentities,approximationsandrelationspresentedherewere
obviouslynotinventedbutcollected,borrowedandcopiedfromalargeamount
ofsources.Thesesourcesincludesimilarbutshorternotesfoundontheinternet
andappendicesinbooks-seethereferencesforafulllist.
Errors:Verylikelythereareerrors,typos,andmistakesforwhichweapolo-
gizeandwouldbegratefultoreceivecorrectionsatcookbook@2302.dk.
Itsongoing:Theprojectofkeepingalargerepositoryofrelationsinvolving
matricesisnaturallyongoingandtheversionwillbeapparentfromthedatein
theheader.
Suggestions:Yoursuggestionforadditionalcontentorelaborationofsome
topicsismostwelcomeatcookbook@2302.dk.
Keywords:Matrixalgebra,matrixrelations,matrixidentities,derivativeof
determinant,derivativeofinversematrix,dierentiateamatrix.
Acknowledgements:Wewouldliketothankthefollowingforcontributions
andsuggestions:BillBaxter,BrianTempleton,ChristianRishøj,Christian
Schr¨oppelDouglasL.Theobald,EsbenHoegh-Rasmussen,GlynneCasteel,Jan
Larsen,JunBinGao,J¨urgenStruckmeier,KamilDedecius,KorbinianStrim-
mer,LarsChristiansen,LarsKaiHansen,LelandWilkinson,LiguoHe,Loic
Thibaut,MiguelBar˜ao,OleWinther,PavelSakov,StephanHattinger,Vasile
Sima,VincentRabaud,ZhaoshuiHe.WewouldalsolikethankTheOticon
FoundationforfundingourPhDstudies.
1
CONTENTS CONTENTS
Contents
1Basics 5
1.1TraceandDeterminants....................... 5
1.2TheSpecialCase2x2......................... 5
2Derivatives 7
2.1DerivativesofaDeterminant.................... 7
2.2DerivativesofanInverse....................... 8
2.3DerivativesofEigenvalues...................... 9
2.4DerivativesofMatrices,VectorsandScalarForms ........ 9
2.5DerivativesofTraces......................... 11
2.6Derivativesofvectornorms..................... 13
2.7Derivativesofmatrixnorms..................... 13
2.8DerivativesofStructuredMatrices................. 14
3Inverses 16
3.1Basic.................................. 16
3.2ExactRelations............................ 17
3.3ImplicationonInverses........................ 19
3.4Approximations............................ 19
3.5GeneralizedInverse.......................... 20
3.6PseudoInverse............................ 20
4ComplexMatrices 23
4.1ComplexDerivatives......................... 23
4.2Higherorderandnon-linearderivatives............... 26
4.3Inverseofcomplexsum....................... 26
5SolutionsandDecompositions 27
5.1Solutionstolinearequations..................... 27
5.2EigenvaluesandEigenvectors.................... 29
5.3SingularValueDecomposition.................... 30
5.4TriangularDecomposition...................... 32
5.5LUdecomposition.......................... 32
5.6LDMdecomposition......................... 32
5.7LDLdecompositions......................... 32
6StatisticsandProbability 33
6.1DefinitionofMoments........................ 33
6.2ExpectationofLinearCombinations................ 34
6.3WeightedScalarVariable ...................... 35
Petersen&Pedersen,TheMatrixCookbook,Version:November14,2008,Page2
CONTENTS CONTENTS
7MultivariateDistributions 36
7.1Cauchy................................ 36
7.2Dirichlet................................ 36
7.3Normal ................................ 36
7.4Normal-InverseGamma....................... 36
7.5Gaussian................................ 36
7.6Multinomial.............................. 36
7.7Student’st .............................. 36
7.8Wishart................................ 37
7.9Wishart,Inverse........................... 38
8Gaussians 39
8.1Basics................................. 39
8.2Moments ............................... 41
8.3Miscellaneous............................. 43
8.4MixtureofGaussians......................... 44
9SpecialMatrices 45
9.1Blockmatrices............................ 45
9.2DiscreteFourierTransformMatrix,The.............. 46
9.3HermitianMatricesandskew-Hermitian.............. 47
9.4IdempotentMatrices......................... 48
9.5Orthogonalmatrices......................... 48
9.6PositiveDefiniteandSemi-definiteMatrices............ 50
9.7SingleentryMatrix,The....................... 51
9.8Symmetric,Skew-symmetric/Antisymmetric............ 53
9.9ToeplitzMatrices........................... 54
9.10Transitionmatrices.......................... 55
9.11Units,PermutationandShift.................... 56
9.12VandermondeMatrices........................ 57
10FunctionsandOperators 58
10.1FunctionsandSeries......................... 58
10.2KroneckerandVecOperator .................... 59
10.3VectorNorms............................. 61
10.4MatrixNorms............................. 61
10.5Rank.................................. 62
10.6IntegralInvolvingDiracDeltaFunctions.............. 62
10.7Miscellaneous............................. 63
AOne-dimensionalResults 64
A.1Gaussian................................ 64
A.2OneDimensionalMixtureofGaussians............... 65
BProofsandDetails 67
B.1MiscProofs.............................. 67
Petersen&Pedersen,TheMatrixCookbook,Version:November14,2008,Page3
CONTENTS CONTENTS
NotationandNomenclature
A Matrix
A
ij
Matrixindexedforsomepurpose
A
i
Matrixindexedforsomepurpose
A
ij
Matrixindexedforsomepurpose
A
n
Matrixindexedforsomepurposeor
Then.thpowerofasquarematrix
A
−1
TheinversematrixofthematrixA
A
+
ThepseudoinversematrixofthematrixA(seeSec.3.6)
A
1/2
Thesquarerootofamatrix(ifunique),notelementwise
(A)
ij
The(i,j).thentryofthematrixA
A
ij
The(i,j).thentryofthematrixA
[A]
ij
Theij-submatrix,i.e.Awithi.throwandj.thcolumndeleted
a Vector
a
i
Vectorindexedforsomepurpose
a
i
Thei.thelementofthevectora
a Scalar
<z Realpartofascalar
<z Realpartofavector
<Z Realpartofamatrix
=z Imaginarypartofascalar
=z Imaginarypartofavector
=Z Imaginarypartofamatrix
det(A)DeterminantofA
Tr(A) TraceofthematrixA
diag(A)DiagonalmatrixofthematrixA,i.e.(diag(A))
ij
=
ij
A
ij
eig(A) EigenvaluesofthematrixA
vec(A)Thevector-versionofthematrixA(seeSec.10.2.2)
sup Supremumofaset
||A|| Matrixnorm(subscriptifanydenoteswhatnorm)
A
T
Transposedmatrix
A
−T
Theinverseofthetransposedandviceversa,A
−T
=(A
−1
)
T
=(A
T
)
−1
.
A
Complexconjugatedmatrix
A
H
Transposedandcomplexconjugatedmatrix(Hermitian)
ABHadamard(elementwise)product
ABKroneckerproduct
0 Thenullmatrix.Zeroinallentries.
I Theidentitymatrix
J
ij
Thesingle-entrymatrix,1at(i,j)andzeroelsewhere
Apositivedefinitematrix
Adiagonalmatrix
Petersen&Pedersen,TheMatrixCookbook,Version:November14,2008,Page4
1BASICS
1Basics
(AB)
−1
=B
−1
A
−1
(1)
(ABC...)
−1
=...C
−1
B
−1
A
−1
(2)
(A
T
)
−1
=(A
−1
)
T
(3)
(A+B)
T
=A
T
+B
T
(4)
(AB)
T
=B
T
A
T
(5)
(ABC...)
T
=...C
T
B
T
A
T
(6)
(A
H
)
−1
=(A
−1
)
H
(7)
(A+B)
H
=A
H
+B
H
(8)
(AB)
H
=B
H
A
H
(9)
(ABC...)
H
=...C
H
B
H
A
H
(10)
1.1TraceandDeterminants
Tr(A)=
P
i
A
ii
(11)
Tr(A)=
P
i
i
,
i
=eig(A) (12)
Tr(A)=Tr(A
T
) (13)
Tr(AB)=Tr(BA) (14)
Tr(A+B)=Tr(A)+Tr(B) (15)
Tr(ABC)=Tr(BCA)=Tr(CAB) (16)
det(A)=
Q
i
i
i
=eig(A) (17)
det(cA)=c
n
det(A), ifA2
R
1.2TheSpecialCase2x2
ConsiderthematrixA
A
11
A
12
A
21
A
22
A=
Determinantandtrace
det(A)=A
11
A
22
−A
12
A
21
(25)
Tr(A)=A
11
+A
22
(26)
Petersen&Pedersen,TheMatrixCookbook,Version:November14,2008,Page5
n×n
(18)
det(A
T
)=det(A) (19)
det(AB)=det(A)det(B) (20)
det(A
−1
)=1/det(A) (21)
det(A
n
)=det(A)
n
(22)
det(I+uv
T
)=1+u
T
v (23)
det(I+"A)
=
1+"Tr(A), "small (24)
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