內(nèi)容分析案例選讀2018the matrix cookbook

1、The Matrix CookbookKaare Brandt Petersen Michael Syskind PedersenVersion:November 15, 20121IntroductionWhat 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 conv

2、enience of anyone who wants a quick desktop reference .Disclaimer: The identities, approximations and relations presented here were obviously not invented but collected, borrowed and copied from a large amount of sources. These sources include similar but shorter notes found on the internet and appe

3、ndices in books - see the references for a full list.Errors: Very likely there are errors, typos, and mistakes for which we apolo- gize and would be grateful to receive corrections at cookbook2302.dk.Its ongoing: The project of keea large repository of relations involvingmatrices is naturally ongoin

4、g and the version will be apparent from the date in the header.Suggestions: Your suggestion for additional content or elaboration of some topics is most welcome acookbook2302.dk.Keywords: Matrix algebra, matrix relations, matrix identities, derivative of determinant, derivative of inverse matrix, di

5、fferentiate a matrix.Acknowledgements: We would like to thank the following for contributions and suggestions: Bill Baxter, Brian Templeton, Christian Rishøj, Christian Schr¨oppel, Dan Boley, Douglas L. Theobald, Esben Hoegh-Rasmussen, Evripidis Karseras, Georg Martius, Glynne Casteel, Jan

6、 Larsen, Jun Bin Gao, Ju¨rgen Struckmeier, Kamil Dedecius, Karim T. Abou-Moustafa, Korbinian Strimmer, Lars Christiansen, Lars Kai Hansen, Leland Wilkinson, Liguo He, Loic Thibaut, Markus Froeb, Michael Hubatka, Miguel Barao, Ole Winther, Pavel Sakov, Stephan Hattinger, Troels Pedersen, Vasile

7、Sima, Vincent Rabaud, Zhaoshui He. We would also like thank The Oticon Foundation for funding our PhD studies.Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 2CONTENTSCONTENTSContents1Basics666Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8、. . . .Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The Special Case 2x2 . . . . . . . . . . . . . . . . . . . . . . . . .2Derivatives88910101214142.72.8Derivatives of a Determinant . . . . . . . . . . . . . . . . . . . .Derivatives of an Inverse . . . .

9、 . . . . . . . . . . . . . . . . . . .Derivatives of Eigenvalues . . . . . . . . . . . . . . . . . . . . . .Derivatives of Matrices, Vectors and Scalar Forms . . . . . . . .Derivatives of Traces . . . . . . . . . . . . . . . . . . . . . . . . .Derivatives of vector norms . . . . . . . . . . . . . .

10、. . . . . . .Derivatives of matrix norms . . . . . . . . . . . . . . . . . . . . .Derivatives of Structured Matrices . . . . . . . . . . . . . . . . .3Inverses171718202021Basic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Exact Relations . . . . . . . . . .

11、. . . . . . . . . . . . . . . . . .Implication on Inverses . . . . . . . . . . . . . . . . . . . . . . . .Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . .Generalized Inverse . . . . . . . . . . . . . . . . . . . . . . . . . .Pseudo Inverse . . . . . . . . . . . . . . . . . . .

12、. . . . . . . . .4Complex Matrices2424262Complex Derivatives . . . . . . . . . . . . . . . . . . . . . . . . .Higher order and non-linear derivatives . . . . . . . . . . . . . . .Inverse of complex sum . . . . . . . . . . . . . . . . . . . . . . .5Solutions and Decompositions28283031323233

13、35.7Solutions to linear equations . . . . . . . . . . . . . . . . . . . . .Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . .Singular Value Decomposition . . . . . . . . . . . . . . . . . . . .Triangular Decomposition . . . . . . . . . . . . . . . . . . . . . .LU

14、 decomposition . . . . . . . . . . . . . . . . . . . . . . . . . .LDM decomposition . . . . . . . . . . . . . . . . . . . . . . . . .LDL decompositions . . . . . . . . . . . . . . . . . . . . . . . . .6Statistics and Probability3434353Definition of Moments . . . . . . . . . . . . . . . . .

15、 . . . . . . .Expectation of Linear Combinations . . . . . . . . . . . . . . . .Weighted Scalar Variable . . . . . . . . . . . . . . . . . . . . . .7Multivariate Distributions3737373737373Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Dirichlet . . . . . . .

16、 . . . . . . . . . . . . . . . . . . . . . . . . .Normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Normal-Inverse Gamma . . . . . . . . . . . . . . . . . . . . . . .Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Multinomial . . . . . . . . . . . . . . .

17、. . . . . . . . . . . . . . .Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 3CONTENTSCONTENTSs t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Wishart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Wishart, Inverse . . . . . . . .

18、 . . . . . . . . . . . . . . . . . . .3738398Gaussians4040424448.4Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Mixture of G

19、aussians . . . . . . . . . . . . . . . . . . . . . . . . .9Special Matrices46464748494950525454555652Block matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . .Discrete Fourier Transform Matrix, The . . . . . . . . . . . . . .Hermitian Matrices and s

20、kew-Hermitian . . . . . . . . . . . . . .Idempotent Matrices . . . . . . . . . . . . . . . . . . . . . . . . .Orthogonal matrices . . . . . . . . . . . . . . . . . . . . . . . . .Positive Definite and Semi-definite Matrices . . . . . . . . . . . .Singleentry Matrix, The . . . . . . . . . . . . . . .

21、 . . . . . . . .Symmetric, Skew-symmetric/symmetric . . . . . . . . . . . .Toeplitz Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . .Transition matrices . . . . . . . . . . . . . . . . . . . . . . . . . .Units, Permutation and Shift . . . . . . . . . . . . . . . . . . . .Vandermonde Mat

22、rices . . . . . . . . . . . . . . . . . . . . . . . .10 Functions and Operators585859616162626310.410.510.610.7Functions and Series . . . . . . . . . . . . . . . . . . . . . . . . .Kronecker and Vec Operator . . . . . . . . . . . . . . . . . . . .Vector Norms . . . . . . . . . . . . . .

23、. . . . . . . . . . . . . . .Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Integral Involving Dirac Delta Functions . . . . . . . . . . . . . .Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . .

24、 . . . . .AOne-dimensional ResultsA.1 Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A.2 One Dimensional Mixture of Gaussians . . . . . . . . . . . . . . .646465BProofs ands6666B.1 Misc Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Petersen & Pedersen

25、, The Matrix Cookbook, Version: November 15, 2012, Page 4CONTENTSCONTENTSNotation and NomenclatureAAij Ai Aij AnMatrixMatrix indexed for some purpose Matrix indexed for some purpose Matrix indexed for some purpose Matrix indexed for some purpose or The n.th power of a square matrix The inverse matri

26、x of the matrix AThe pseudo inverse matrix of the matrix A (see Sec. 3.6) The square root of a matrix (if unique), not elementwise The (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 deleted Vector (column-vector)Vector indexed

27、 for some purpose The i.th element of the vector a ScalarA1A+ A1/2(A)ij Aij Aij aai ai a<zReal part of a scalar Real part of a vector Real part of a matrixImaginary part of a scalar Imaginary part of a vector Imaginary part of a matrix<z<Z=z=z=Zdet(A)Tr(A)diag(A) eig(A)vec(A) supDeterminant

28、 of ATrace of the matrix ADiagonal matrix of the matrix A, i.e. (diag(A)ij = ijAijEigenvalues of the matrix AThe vector-version of the matrix A (see Sec. 10.2.2) Supremum of a setMatrix norm (subscript if any denotes what norm) Transposed matrixThe inverse of the transposed and vice versa, AT = (A1)

29、T = (AT )1.Complex conjugated matrixTransposed and complex conjugated matrix (Hermitian)|A|AT ATAAHA BHadamard (elementwise) product Kronecker productA B0IJij The null matrix. Zero in all entries. The identity matrixThe single-entry matrix, 1 at (i, j) and zero elsewhere A positive definite matrixA

30、diagonal matrixPetersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 51BASICS1Basics(AB)1 (ABC.)1(AT )1(A + B)T(AB)T (ABC.)T (AH )1(A + B)H(AB)H (ABC.)HB1A1.C1B1A1(A1)T AT + BT BT AT.CT BT AT (A1)H AH + BH BH AH.CH BH AH=(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)1.1TraceTr(A)Tr(A)Tr(A)

31、Tr(AB) Tr(A + B) Tr(ABC)aT a=A(11)(12)(13)(14)(15)(16)(17)iiiPii,i = eig(A)Tr(AT )Tr(BA)Tr(A) + Tr(B)Tr(BCA) = Tr(CAB)Tr(aaT )1.2DeterminantLet A be an n × n matrix.det(A) det(cA) det(AT ) det(AB) det(A1) det(An) det(I + uvT )Q=i = eig(A)(18)(19)(20)(21)(22)(23)(24)iicn det(A), det(A) det(A) de

32、t(B) 1/ det(A) det(A)n1 + uT vif A Rn×nFor n = 2:det(I + A) = 1 + det(A) + Tr(A)(25)For n = 3:1122det(I + A) = 1 + det(A) + Tr(A) + Tr(A) Tr(A )22(26)Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 61.3 The Special Case 2x21BASICSFor n = 4:12det(I + A) = 1 + det(A

33、) + Tr(A) +122+Tr(A) Tr(A )2111323+ Tr(A) Tr(A)Tr(A ) + Tr(A ) 623For small , the following approximation holds(27)112222det(I + A) = 1 + det(A) + Tr(A) + Tr(A) Tr(A )(28)221.3The Special Case 2x2Consider the matrix AA = A11 A21A12 A22Determinant and tracedet(A) = A11A22 A12A21Tr(A) = A11 + A22(29)(

34、30)Eigenvalues2 · Tr(A) + det(A) = 0pp22Tr(A) +Tr(A) 4 det(A)Tr(A) Tr(A) 4 det(A)1 =2 =21 + 2 = Tr(A)212 = det(A)Eigenvectorsv1 v2 A12 1 A11A12 2 A11Inverse 1A22A21A121A=(31)A11det(A)Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 72 DERIVATIVES2DerivativesThis se

35、ction is covering differentiation of a number of expressions with respect to a 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, positive definite). See section 2.8 for differentiation of structured ma

36、trices. The basic assumptions can be written in a formula asXklXij= (32)ik ljthat is for e.g. vector forms, xxi=xyxyixyxiyj=yyiiijThe following rules are general and very useful when deriving the differential of an expression (19):A(X)(X + Y)(Tr(X)(XY)=0XX + YTr(X)(X)Y + X(Y)(A is a constant)(33)(34

37、)(35)(36)(37)(38)(39)(40)(41)(42)(43)(44)(45)(X Y)(X) Y + X (Y)(X Y)(X) Y + X (Y)(X1)(det(X)(det(X)(ln(det(X)XTXHX1(X)X1Tr(adj(X)X) det(X)Tr(X1X) Tr(X1X)(X)T(X)H2.1Derivativesof a Determinant2.1.1General form det(Y)xYx1=det(Y)Tr Y(46)X det(X)XikX= det(X)(47)jkijk""#1Y2 det(Y)1 x=det(Y) Tr

38、Yx2x YxYxYx1+Tr YTr Y# Yx11TrYY(48)Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 82.2 Derivatives of an Inverse2DERIVATIVES2.1.2Linear forms det(X)det(X)(X1)T=(49)X det(X)XX= det(X)(50)jkijXikk det(AXB)det(AXB)(X1)T = det(AXB)(XT )1=(51)X2.1.3Square formsIf X is squa

39、re and invertible, then det(XT AX)TT= 2 det(X AX)X(52)XIf X is not square but A is symmetric, then det(XT AX)TT1= 2 det(X AX)AX(X AX)(53)XIf X is not square and A is not symmetric, then det(XT AX)TT1TTT1= det(X AX)(AX(X AX)+ A X(X A X)(54)X2.1.4Other nonlinear formsSome special cases are (See 9, 7)

40、ln det(XT X)|+ T=2(X )(55)X ln det(XT X)T=2X(56)X+ ln | det(X)|(X1)T= (XT )1=(57)X det(Xk)kT=k det(X )X(58)X2.2Derivatives of an InverseFrom 27 we have the basic identityY11 Y1= YYx(59)xPetersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 92.3 Derivatives of Eigenvalues2DER

41、IVATIVESfrom which it follows(X1)kl11(X)ki(X=)jl(60)XijaT X1bTTT=Xab X(61)X det(X1)11 T= det(X)(X)(XBAX)(62)XTr(AX1B)11 T=(63)XTr(X + A)1)11 T=(X + A)(X + A)(64)XFrom 32 we have the following result: Let A be an n × n invertible squarematrix, W be the inverse of A, and J (A) is an n × n -v

42、ariate and differentiablefunction with respect to A, then the partial differentials of J with respect to Aand W satisfyJAJW= ATAT2.3Derivatives of EigenvaluesX eig(X) X X X X=Tr(X)=I(65)Y eig(X)=det(X)XT=det(X)(66)If A is real and symmetric, i and vi are distinct eigenvalues and eigenvectorsof A (se

43、e (276) with vT vi = 1, then 33ii= vT (A)vi(67)(68)i+vi= (iI A) (A)vi2.4 Derivatives of Matrices, Vectors and Scalar2.4.1 First OrderFormsxT aaT x=a(69)xaT XbxT=ab(70)XaT XT bT=ba(71)XaT XaaT XT aT=aa(72)XXXJij=(73)Xij(XA)ijXmn(XT A)ij(JmnA)=(A)=(74)imnjijnm=in(A)mj=(JA)ij(75)XmnPetersen & Peder

44、sen, The Matrix Cookbook, Version: November 15, 2012, Page 102.4 Derivatives of Matrices, Vectors and Scalar Forms2DERIVATIVES2.4.2Second OrderXX XijX X=2X(76)kl mnklklmnklbT XT XcTT=X(bc + cb )(77)X(Bx + b)T C(Dx + d)TTT=B C(Dx + d) + D C (Bx + b)(78)x(XT BX)klT=lj(X B)ki + kj(BX)il(79)Xij(XT BX)Ti

45、jjiij=X BJ + J BX(J)kl = ikjl(80)XijSee Sec 9.7 for useful properties of the Single-entry matrix JijxT BxT=(B + B )x(81)xbT XT DXcTTT=D Xbc + DXcb(82)X(Xb + c)T D(Xb + c)X(D + DT )(Xb + c)bT=(83)Assume W is symmetric, thensT(x As) W(x As)T2A W(x As)2W(x s)2W(x s) 2W(x As)T2W(x As)s=(84)xsT(x s) W(x

46、s)T(x s) W(x s)=(85)=(86)x AT(x As) W(x As)T(x As) W(x As)=(87)=(88)As a case with complex values the following holds(a xH b)2H= 2b(a x b)(89)xThis formula is also known from the LMS algorithm 142.4.3Higher-order and non-linearn1Xn(X )klXijr ij n1r=(X J X)(90)klr=0For proof of the above, see B.1.3.n

47、1X Xn1r TTnr TTa X b =(X ) ab (X)(91)r=0Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 112.5 Derivatives of Traces2 DERIVATIVEShn1X XXn1rabT (Xn)T XraT (Xn)T Xnb =r=0+(Xr)T XnabT (Xn1r)T i(92)See B.1.3 for a proof.Assume s and r are functions of x, i.e. s = s(x), r =

48、r(x), and that A is a constant, then TTxsxrxsT ArAT s=Ar +(93) (Ax)T (Ax) xT AT Ax=(94)x (Bx)T (Bx)x xT BT BxAT AxxT AT AxBT Bx=2 2xT BBx(xT BT Bx)2(95)2.4.4Gradient and HessianUsing the above we have for the gradient and the HessianxT Ax + bT x(A + A )x + bf=(96)fTxf = x2f=(97)T=A + A(98)xxT2.5Derivatives of TracesAssume F (X) to be a differentiable function of each of the elements of X. It then holds thatTr(F (X)= f (X)TXwhere f (·) is the scalar derivative of F (·).2.5.1First OrderXTr(X)=I(99)ATTr(XA)=(100)XAT BTTr(AXB)=(101)XT