The Matrix Cookbook
The Matrix CookbookKaare Brandt PetersenMichael Syskind PedersenVersion: February 16, 2006What is this? These pages are a collection of facts (identities, approximations, inequalities, relations, .) about matrices and matters relating to them.It is collected in this form for the convenience of anyone who wants a quickdesktop reference .Disclaimer: The identities, approximations and relations presented here wereobviously not invented but collected, borrowed and copied from a large amountof sources. These sources include similar but shorter notes found on the internetand appendices in books - see the references for a full list.Errors: Very likely there are errors, typos, and mistakes for which we apologize and would be grateful to receive corrections at cookbook@2302.dk.Its ongoing: The project of keeping a large repository of relations involvingmatrices is naturally ongoing and the version will be apparent from the date inthe header.Suggestions: Your suggestion for additional content or elaboration of sometopics is most welcome at cookbook@2302.dk.Keywords: Matrix algebra, matrix relations, matrix identities, derivative ofdeterminant, derivative of inverse matrix, differentiate a matrix.Acknowledgements: We would like to thank the following for contributions and suggestions: Christian Rishøj, Douglas L. Theobald, Esben HoeghRasmussen, Lars Christiansen, and Vasile Sima. We would also like thank TheOticon Foundation for funding our PhD studies.1
CONTENTSCONTENTSContents1 Basics1.1 Trace and Determinants . . . . . . . . . . . . . . . . . . . . . . .1.2 The Special Case 2x2 . . . . . . . . . . . . . . . . . . . . . . . . .2 Derivatives2.1 Derivatives2.2 Derivatives2.3 Derivatives2.4 Derivatives2.5 Derivativesofofofofofa Determinant . . . . . . . . . . . .an Inverse . . . . . . . . . . . . . . .Matrices, Vectors and Scalar FormsTraces . . . . . . . . . . . . . . . . .Structured Matrices . . . . . . . . .3 Inverses3.1 Basic . . . . . . . . . . .3.2 Exact Relations . . . . .3.3 Implication on Inverses .3.4 Approximations . . . . .3.5 Generalized Inverse . . .3.6 Pseudo Inverse . . . . .555.77891112.151516171717174 Complex Matrices194.1 Complex Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 195 Decompositions225.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . 225.2 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . 225.3 Triangular Decomposition . . . . . . . . . . . . . . . . . . . . . . 246 Statistics and Probability256.1 Definition of Moments . . . . . . . . . . . . . . . . . . . . . . . . 256.2 Expectation of Linear Combinations . . . . . . . . . . . . . . . . 266.3 Weighted Scalar Variable . . . . . . . . . . . . . . . . . . . . . . 277 Gaussians7.1 Basics . . . . . . . .7.2 Moments . . . . . .7.3 Miscellaneous . . . .7.4 Mixture of Gaussians.28283032338 Special Matrices8.1 Units, Permutation and Shift .8.2 The Singleentry Matrix . . . .8.3 Symmetric and Antisymmetric8.4 Vandermonde Matrices . . . . .8.5 Toeplitz Matrices . . . . . . . .8.6 The DFT Matrix . . . . . . . .34343537373839.Petersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 2
CONTENTS8.78.8CONTENTSPositive Definite and Semi-definite Matrices . . . . . . . . . . . .Block matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 Functions and Operators9.1 Functions and Series . . . . . . . . . . .9.2 Kronecker and Vec Operator . . . . . .9.3 Solutions to Systems of Equations . . .9.4 Matrix Norms . . . . . . . . . . . . . . .9.5 Rank . . . . . . . . . . . . . . . . . . . .9.6 Integral Involving Dirac Delta Functions9.7 Miscellaneous . . . . . . . . . . . . . . .40414343444547484849A One-dimensional Results50A.1 Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50A.2 One Dimensional Mixture of Gaussians . . . . . . . . . . . . . . . 51B Proofs and Details53B.1 Misc Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Petersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 3
CONTENTSCONTENTSNotation and NomenclatureAAijAiAijAnA 1A A1/2(A)ijAij[A]ijaaiaia z z Z z z ZMatrixMatrix indexed for some purposeMatrix indexed for some purposeMatrix indexed for some purposeMatrix indexed for some purpose orThe n.th power of a square matrixThe inverse matrix of the matrix AThe pseudo inverse matrix of the matrix A (see Sec. 3.6)The square root of a matrix (if unique), not elementwiseThe (i, j).th entry of the matrix AThe (i, j).th entry of the matrix AThe ij-submatrix, i.e. A with i.th row and j.th column deletedVectorVector indexed for some purposeThe i.th element of the vector aScalarReal part of a scalarReal part of a vectorReal part of a matrixImaginary part of a scalarImaginary part of a vectorImaginary part of a matrixdet(A)Tr(A)diag(A)vec(A) A ATA AHDeterminant of ATrace of the matrix ADiagonal matrix of the matrix A, i.e. (diag(A))ij δij AijThe vector-version of the matrix A (see Sec. 9.2.2)Matrix norm (subscript if any denotes what norm)Transposed matrixComplex conjugated matrixTransposed and complex conjugated matrix (Hermitian)A BA BHadamard (elementwise) productKronecker product0IJijΣΛThe null matrix. Zero in all entries.The identity matrixThe single-entry matrix, 1 at (i, j) and zero elsewhereA positive definite matrixA diagonal matrixPetersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 4
1 BASICS1Basics(AB) 1(ABC.) 1(AT ) 1(A B)T(AB)T(ABC.)T(AH ) 1(A B)H(AB)H(ABC.)H1.1B 1 A 1.C 1 B 1 A 1(A 1 )TAT B TB T AT.CT BT AT(A 1 )HAH B HB H AH.CH BH AHTrace and DeterminantsTr(A)Tr(A)Tr(A)Tr(AB)Tr(A B)Tr(ABC)det(A)det(AB)det(A 1 )det(I uvT )1.2 PAiiPiλi eig(A)i λi ,TTr(A )Tr(BA)Tr(A) Tr(B)Tr(BCA) Tr(CAB)Qλi eig(A)i λidet(A) det(B)1/ det(A)1 uT vThe Special Case 2x2Consider the matrix A A A11A21A12A22 Determinant and tracedet(A) A11 A22 A12 A21Tr(A) A11 A22Eigenvaluesλ2 λ · Tr(A) det(A) 0Petersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 5
1.2The Special Case 2x2λ1 Tr(A) 1pTr(A)2 4 det(A)2λ1 λ2 Tr(A)Eigenvectors v1 A12λ1 A11InverseA 1λ2 pTr(A)2 4 det(A)2λ1 λ2 det(A) 1 det(A)Tr(A) BASICS v2 A22 A21A12λ2 A11 A12A11 Petersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 6
2 DERIVATIVES2DerivativesThis section is covering differentiation of a number of expressions with respect toa matrix X. Note that it is always assumed that X has no special structure, i.e.that the elements of X are independent (e.g. not symmetric, Toeplitz, positivedefinite). See section 2.5 for differentiation of structured matrices. The basicassumptions can be written in a formula as Xkl δik δlj Xijthat is for e.g. vector forms, x xi y i y x y i x yi x y ij xi yjThe following rules are general and very useful when deriving the differential ofan expression ([13]): A (αX) (X Y) (Tr(X)) (XY) (X Y) (X Y) (X 1 ) (det(X)) (ln(det(X))) XT XH2.12.1.1 0(A is a constant)α X X YTr( X)( X)Y X( Y)( X) Y X ( Y)( X) Y X ( Y) X 1 ( X)X 1det(X)Tr(X 1 X)Tr(X 1 X)( X)T( ves of a DeterminantGeneral form det(Y) Y det(Y)Tr Y 1 x x2.1.2Linear forms det(X) det(X)(X 1 )T X det(AXB) det(AXB)(X 1 )T det(AXB)(XT ) 1 XPetersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 7
2.2Derivatives of an Inverse2.1.32DERIVATIVESSquare formsIf X is square and invertible, then det(XT AX) 2 det(XT AX)X T XIf X is not square but A is symmetric, then det(XT AX) 2 det(XT AX)AX(XT AX) 1 XIf X is not square and A is not symmetric, then det(XT AX) det(XT AX)(AX(XT AX) 1 AT X(XT AT X) 1 ) X2.1.4(13)Other nonlinear formsSome special cases are (See [8, 7]) ln det(XT X) X ln det(XT X) X ln det(X) X det(Xk ) X2.2 2(X )T 2XT (X 1 )T (XT ) 1 k det(Xk )X TDerivatives of an InverseFrom [19] we have the basic identity Y 1 Y 1 Y 1Y x xfrom which it follows (X 1 )kl Xij aT X 1 b X det(X 1 ) X Tr(AX 1 B) X (X 1 )ki (X 1 )jl X T abT X T det(X 1 )(X 1 )T (X 1 BAX 1 )TPetersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 8
2.3Derivatives of Matrices, Vectors and Scalar Forms2.32.3.1DERIVATIVESDerivatives of Matrices, Vectors and Scalar FormsFirst Order xT a x aT Xb X aT XT b X aT Xa X X Xij (XA)ij Xmn (XT A)ij Xmn2.3.22 aT x x a abT baT aT XT a X aaT δim (A)nj (Jmn A)ij δin (A)mj (Jnm A)ij JijSecond Order XXkl Xmn Xij 2X bT XT Xc X (Bx b)T C(Dx d) x (XT BX)kl Xij (XT BX) XijXklklklmn X(bcT cbT ) BT C(Dx d) DT CT (Bx b) δlj (XT B)ki δkj (BX)il XT BJij Jji BX(Jij )kl δik δjlSee Sec 8.2 for useful properties of the Single-entry matrix Jij xT Bx x bT XT DXc X (Xb c)T D(Xb c) X (B BT )x DT XbcT DXcbT (D DT )(Xb c)bTPetersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 9
2.3Derivatives of Matrices, Vectors and Scalar Forms2DERIVATIVESAssume W is symmetric, then (x As)T W(x As) s (x s)T W(x s) s (x As)T W(x As) x (x As)T W(x As) A2.3.3 2AT W(x As) 2W(x s) 2W(x As) 2W(x As)sTHigher order and non-linearn 1X T na X b (Xr )T abT (Xn 1 r )T Xr 0 T n T na (X ) X b X(14)n 1XhXn 1 r abT (Xn )T Xrr 0 (Xr )T Xn abT (Xn 1 r )Ti(15)See B.1.1 for a proof.Assume s and r are functions of x, i.e. s s(x), r r(x), and that A is aconstant, then Ts Ar x2.3.4 s x T Ar sT A r x Gradient and HessianUsing the above we have for the gradient and the hessianf f x f x 2f x xT xT Ax bT x (A AT )x b A ATPetersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 10
2.4Derivatives of Traces2.42DERIVATIVESDerivatives of Traces2.4.1First Order Tr(X) X Tr(XA) X Tr(AXB) X Tr(AXT B) X Tr(XT A) X Tr(AXT ) X2.4.2 I AT AT B T BA A A(16)Second Order Tr(X2 ) 2XT X Tr(X2 B) (XB BX)T X Tr(XT BX) X Tr(XBXT ) X Tr(AXBX) X Tr(XT X) X Tr(BXXT ) X Tr(BT XT CXB) X Tr XT BXC X Tr(AXBXT C) Xih Tr (AXb c)(AXb c)T X BX BT X XBT XB AT XT BT BT XT AT 2X (B BT )X CT XBBT CXBBT BXC BT XCT AT CT XBT CAXB 2AT (AXb c)bTSee [7].Petersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 11
2.5Derivatives of Structured Matrices2.4.32DERIVATIVESHigher Order Tr(Xk ) X k(Xk 1 )Tk 1X Tr(AXk ) (Xr AXk r 1 )T Xr 0 T T TT TrBXCXXCXB CXXCXBBT X CT XBBT XT CT X CXBBT XT CX CT XXT CT XBBT2.4.4Other Tr(AX 1 B) (X 1 BAX 1 )T X T AT BT X T XAssume B and C to be symmetric, thenhi Tr (XT CX) 1 A (CX(XT CX) 1 )(A AT )(XT CX) 1 Xhi Tr (XT CX) 1 (XT BX) 2CX(XT CX) 1 XT BX(XT CX) 1 X 2BX(XT CX) 1See [7].2.5Derivatives of Structured MatricesAssume that the matrix A has some structure, i.e. symmetric, toeplitz, etc.In that case the derivatives of the previous section does not apply in general.Instead, consider the following general rule for differentiating a scalar functionf (A)" # TX f Akldf f A TrdAij Akl Aij A AijklThe matrix differentiated with respect to itself is in this document referred toas the structure matrix of A and is defined simply by A Sij AijIf A has no special structure we have simply Sij Jij , that is, the structurematrix is simply the singleentry matrix. Many structures have a representationin singleentry matrices, see Sec. 8.2.6 for more examples of structure matrices.Petersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 12
2.5Derivatives of Structured Matrices2.5.12DERIVATIVESThe Chain RuleSometimes the objective is to find the derivative of a matrix which is a functionof another matrix. Let U f (X), the goal is to find the derivative of thefunction g(U) with respect to X: g(U) g(f (X)) X X(17)Then the Chain Rule can then be written the following way:MN g(U) X X g(U) ukl g(U) X xij ukl xij(18)k 1 l 1Using matrix notation, this can be written as:h g(U) U i g(U) Tr ()T. Xij U Xij2.5.2(19)SymmetricIf A is symmetric, then Sij Jij Jji Jij Jij and therefore T df f f f diagdA A A AThat is, e.g., ([5], [20]): Tr(AX) X det(X) X ln det(X) X2.5.3 A AT (A I), see (23)(20) det(X)(2X 1 (X 1 I))(21) 2X 1 (X 1 I)(22)DiagonalIf X is diagonal, then ([13]): Tr(AX) X2.5.4 A I(23)ToeplitzLike symmetric matrices and diagonal matrices also Toeplitz matrices has aspecial structure which should be taken into account when the derivative withrespect to a matrix with Toeplitz structure.Petersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 13
2.5Derivatives of Structured Matrices Tr(AT) T Tr(TA) TDERIVATIVES(24)Tr([AT ]n1 )Tr(A) 2Tr([AT ]1n ))Tr([[AT ]1n ]n 1,2 ).Tr(A)Tr([[AT ]1n ]2,n 1 ).A1n.···.···.Tr([[AT ]1n ]2,n 1 ).Tr([AT ]1n )) An1.Tr([[AT ]1n ]n 1,2 )Tr([AT ]n1 )Tr(A) α(A)As it can be seen, the derivative α(A) also has a Toeplitz structure. Each valuein the diagonal is the sum of all the diagonal valued in A, the values in thediagonals next to the main diagonal equal the sum of the diagonal next to themain diagonal in AT . This result is only valid for the unconstrained Toeplitzmatrix. If the Toeplitz matrix also is symmetric, the same derivative yields Tr(AT) T Tr(TA) T α(A) α(A)T α(A) I (25)Petersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 14
3 INVERSES33.13.1.1InversesBasicDefinitionThe inverse A 1 of a matrix A Cn n is defined such thatAA 1 A 1 A I,(26)where I is the n n identity matrix. If A 1 exists, A is said to be nonsingular.Otherwise, A is said to be singular (see e.g. [9]).3.1.2Cofactors and AdjointThe submatrix of a matrix A, denoted by [A]ij is a (n 1) (n 1) matrixobtained by deleting the ith row and the jth column of A. The (i, j) cofactorof a matrix is defined ascof(A, i, j) ( 1)i j det([A]ij ),The matrix of cofactors can be created from the cofactors cof(A, 1, 1)···cof(A, 1, n) .cof(A) cof(A,i,j) cof(A, n, 1)···cof(A, n, n)(27)(28)The adjoint matrix is the transpose of the cofactor matrixadj(A) (cof(A))T ,3.1.3(29)DeterminantThe determinant of a matrix A Cn n is defined as (see [9])det(A) nX( 1)j 1 A1j det ([A]1j )j 1 nXA1j cof(A, 1, j).(30)j 13.1.4ConstructionThe inverse matrix can be constructed, using the adjoint matrix, byA 1 1· adj(A)det(A)(31)Petersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 15
3.2Exact Relations3.1.53INVERSESCondition numberThe condition number of a matrix c(A) is the ratio between the largest and thesmallest singular value of a matrix (see Section 5.2 on singular values),c(A) d d The condition number can be used to measure how singular a matrix is. If thecondition number is large, it indicates that the matrix is nearly singular. Thecondition number can also be estimated from the matrix norms. Herec(A) kAk · kA 1 k,(32)where k · k is a norm such as e.g the 1-norm, the 2-norm, the -norm or theFrobenius norm (see Sec 9.4 for more on matrix norms).3.23.2.1Exact RelationsThe Woodbury identity(A CBCT ) 1 A 1 A 1 C(B 1 CT A 1 C) 1 CT A 1If P, R are positive definite, then (see [22])(P 1 BT R 1 B) 1 BT R 1 PBT (BPBT R) 13.2.2The Kailath Variant(A BC) 1 A 1 A 1 B(I CA 1 B) 1 CA 1See [4] page 153.3.2.3The Searle Set of IdentitiesThe following set of identities, can be found in [17], page 151,(I A 1 ) 1 A(A I) 1(A BBT ) 1 B A 1 B(I BT A 1 B) 1(A 1 B 1 ) 1 A(A B) 1 B B(A B) 1 AA A(A B) 1 A B B(A B) 1 BA 1 B 1 A 1 (A B)B 1(I AB) 1 I A(I BA) 1 B(I AB) 1 A A(I BA) 1Petersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 16
3.3Implication on Inverses3.33INVERSESImplication on Inverses(A B) 1 A 1 B 1AB 1 A BA 1 B See [17].3.3.1A PosDef identityAssume P, R to be positive definite and invertible, then(P 1 BT R 1 B) 1 BT R 1 PBT (BPBT R) 1See [22].3.4Approximations(I A) 1 I A A2 A3 .A A(I A) 1 A I A 1if A large and symmetricIf σ 2 is small then(Q σ 2 M) 1 Q 1 σ 2 Q 1 MQ 13.53.5.1Generalized InverseDefinitionA generalized inverse matrix of the matrix A is any matrix A such that (see[18])AA A AThe matrix A is not unique.3.63.6.1Pseudo InverseDefinitionThe pseudo inverse (or Moore-Penrose inverse) of a matrix A is the matrix A that fulfilsIIIIIIAA A AA AA A AA symmetricIVA A symmetricThe matrix A is unique and does always exist.Petersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 17
3.6Pseudo Inverse3.6.23INVERSESPropertiesAssume A to be the pseudo-inverse of A, then (See [3])(A ) (AT ) A (A )T(cA) (1/c)A A (AT ) (AT ) A T (A A)(AAT ) Assume A to have full rank, then(AA )(AA )(A A)(A A)Tr(AA )Tr(A A)3.6.3 AA A A rank(AA ) rank(A A)(See [18])(See [18])ConstructionAssume that A has full rank, thenA n nA n mA n mSquareBroadTallrank(A) nrank(A) nrank(A) m A A 1A AT (AAT ) 1A (AT A) 1 ATAssume A does not have full rank, i.e. A is n m and rank(A) r min(n, m).The pseudo inverse A can be constructed from the singular value decomposition A UDVT , byA VD UTA different way is this: There does always exists two matrices C n r and Dr m of rank r, such that A CD. Using these matrices it holds thatA DT (DDT ) 1 (CT C) 1 CTSee [3].Petersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 18
44COMPLEX MATRICESComplex Matrices4.1Complex DerivativesIn order to differentiate an expression f (z) with respect to a complex z, theCauchy-Riemann equations have to be satisfied ([7]): (f (z)) (f (z))df (z) idz z z(33)anddf (z) (f (z)) (f (z)) i dz z zor in a more compact form: f (z) f (z) i. z z(34)(35)A complex function that satisfies the Cauchy-Riemann equations for points in aregion R is said yo be analytic in this region R. In general, expressions involvingcomplex conjugate or conjugate transpose do not satisfy the Cauchy-Riemannequations. In order to avoid this problem, a more generalized definition ofcomplex derivative is used ([16], [6]): Generalized Complex Derivative:df (z)1 f (z) f (z) i.dz2 z z(36) Conjugate Complex Derivativedf (z)1 f (z) f (z) i.dz 2 z z(37)The Generalized Complex Derivative equals the normal derivative, when f is ananalytic function. For a non-analytic function such as f (z) z , the derivativeequals zero. The Conjugate Complex Derivative equals zero, when f is ananalytic function. The Conjugate Complex Derivative has e.g been used by [14]when deriving a complex gradient.Notice:df (z) f (z) f (z)6 i.(38)dz z z Complex Gradient Vector: If f is a real function of a complex vector z,then the complex gradient vector is given by ([11, p. 798]) f (z) df (z)dz f (z) f (z) i. z z2(39)Petersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 19
4.1Complex Derivatives4COMPLEX MATRICES Complex Gradient Matrix: If f is a real function of a complex matrix Z,then the complex gradient matrix is given by ([2]) f (Z) df (Z)dZ f (Z) f (Z) i. Z Z2(40)These expressions can be used for gradient descent algorithms.4.1.1The Chain Rule for complex numbersThe chain rule is a little more complicated when the function of a complexu f (x) is non-analytic. For a non-analytic function, the following chain rulecan be applied ([7]) g(u) x g u g u u x u x g u g u u x u x(41)Notice, if the function is analytic, the second term reduces to zero, and the fu
The Matrix Cookbook Kaare Brandt Petersen Michael Syskind Pedersen Version: February 16, 2006 What is this? These pages are a collection of facts (identities, approxima-tions, inequalities, relations, .) about matrices and matters relating to them. It is collected in this form for the convenience of anyone who wants a quick desktop reference .
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CONTENTS CONTENTS Notation and Nomenclature A Matrix A ij Matrix indexed for some purpose A i Matrix indexed for some purpose Aij Matrix indexed for some purpose An Matrix indexed for some purpose or The n.th power of a square matrix A 1 The inverse matrix of the matrix A A The pseudo inverse matrix of the matrix A (see Sec. 3.6) A1 2 The square root of a matrix (if unique), not elementwise
A Matrix A ij Matrix indexed for some purpose A i Matrix indexed for some purpose Aij Matrix indexed for some purpose An Matrix indexed for some purpose or The n.th power of a square matrix A 1 The inverse matrix of the matrix A A The pseudo inverse matrix of the matrix A (see Sec. 3.6) A1/2 The square root of a matrix (if unique), not .
CONTENTS CONTENTS Notation and Nomenclature A Matrix Aij Matrix indexed for some purpose Ai Matrix indexed for some purpose Aij Matrix indexed for some purpose An Matrix indexed for some purpose or The n.th power of a square matrix A 1 The inverse matrix of the matrix A A The pseudo inverse matrix of the matrix A (see Sec. 3.6) A1/2 The square root of a matrix (if unique), not elementwise
