矩阵公式大全

【实例简介】
矩阵公式大全，上面有研究生阶段用到的很多矩阵公式，便于查阅。
CONTENTS CONTENTS Contents 1 Basics 1.3 Thc Spccial Casc 2x2. 2 Derivatives 2.1 Derivatives of a determinant 6667889 2.2 Derivatives of an Inverse 2.3 Derivatives of eigenvalues 10 2.4 Derivatives of Matrices, Vectors and Scalar Forms 2.5 Derivatives of Traces 12 2.6 Derivatives of vector norms 14 2. 8 Derivatives of Structured Matrices 3 Inverses 17 3.1 Basic 3.2 Exact Relations 18 3.3 Inplication un inverse 20 3.4 Approximations 20 3.5 Generalized Inverse 21 3.6 Pscudo inverse 4 Conlplex Matrices 24 4.1 Complex Derivatives ,24 4.2 Higher order and non-linear derivatives 26 4. 3 Inverse of complex sum 5 Solutions and Decompositions 5.1 Solutions to linear equations 5.2 Eigenvalues and Eigenvectors.... 30 5. 3 Singular valuc Dccomposition 5.4 Triangular De osition 2 5.5 lu decomposition 5.6 tdM decomposition 5.7 LDl decomposition 33 6 Statistics and Probability 34 6.1 Definition of moments 234 6.2 Expectation of Linear Combinations 35 6. 3 Weighted Scalar Variable 36 7 Multivariatc Distributions 37 37 7.2 Dirichlet 37 7.3 Normal 37 Normal-Inverse gamma .,..37 7.5 Gaussian 37 7.6 Multinomial 37 peteRsen pedersen, THe Matrix CoOKBOOK, version: NoVEMBER 15, 2012, Page 3 CONTENTS CONTENTS 17.7 Students t 7. 8 Wishart 38 7.9 Wishart. Inverse 39 8 Gaussians 40 8.1 Basics 40 8.2 Moments 42 8. 3 Miscellaneous 44 18.4 Mixture of Gaussians 9 Spccial Matrices 46 ⑨, Block matrices 46 9.2 Discrete Fourier Transform Matrix, The 47 9.3 Ilermitian Matrices and skew-Ilermitianl 48 9. 4 Idempotent Matrices 49 .5 Orthogonal illatrice 19.6 Positive Definite and Semi-definite Matrices 50 9.7 Singleentry Matrix, Thel 52 9.8 Symmctric, Skcw-symmctric/ Antisymmetric 9.9 Toeplitz Matrices 54 9.10 Transition matrices 55 9.11 Units, Permutation and Shift 6 9.12 Vandermonde Matrices 10 Functions and Operators 10.1 Functions and Series 10.2 Kronecker and Vec Operator 59 110.3 Vector NorIlIS 10.4 Matrix 10.5 Rank 10.6 Integral Involving Dirac Delta Functions 10.7 Miscellanous 63 A One-diInelsional Results 64 A 1 Gaussian 64 A. 2 One Dimensional Mixture of Gaussians 65 B Proofs and Details 66 B. 1 Misc Proofs 66 peteRsen pedersen, THe Matrix CoOKBOOK, version: NoVEMBER 15, 2012, Page 4 CONTENTS CONTENTS Notation and nomenclature Matrix A Matrix indexed for some purpose A Matrix indexed for some purpose Matrix indexed for some purpose A Matrix indexed for some purpose or The n th power of a The inverse matrix of the matrix a A The pseudo inverse matrix of the matrix A(see Sec. 3.6 A/2 Thc squarc root of a matrix(if unique), not clcmcntwisc (A)ij The (i,).th entry of the matrix A A The(i,]). th entry of the matrix A Aij The ij-submatrix, i.e. A with i th row and j th column deleted Vector(column-vector) Vector indexed for solnle purpose The i th element of the vector a Scalar 9 Real part of lar 9 Real part of a vecto 成 z Real part of a matrix 之 Imaginary part of a scalar Sz Imaginary part of a vector SZ Imaginary part of a matrix det(a) Determinant of A Tr(A) Trace of the matrix A (A) Diagonal matrix of the matrix A, i.e.(diag(A))ii-8ij Aij A) Eigenvalues of the matrix A A)The vector-version of the Matrix A(see Sec. 10.2. sup Supremum of a set IA Matrix norm(subscript if any denotes what norm) A Transposed matrix A- The inverse of the transposed and vice versa, A=(A A Complex coj A Transposed and complex conjugated matrix(Hermitian) A oB Hadamard (elementwise) product A⑧ B Kronecker product 0 The null matrix, Zero in all entries JU The single-entry matrix, 1 at(i,]) and zero elsewhere a positive definite matrix A a diagonal matrix peteRsen pedersen, THe Matrix CoOKBOOK, version: NoVEMBER 15, 2012, Page 5 BASICS 1 Basics (AB)= BA CABC CBA (A 1\T (A-B)=A+B aB BA (5) (ABC CBA 6) A 1\H (A+B a+B (8 (AB) B A (ABC. C HRH B A H (10) 1.1Tr Tr(A) Tr(A) (A (12) (A) Ir(a (13) Tr(AB)- Tr(BA) Tr(A+B) Tr (A)+Tr(B) 15) Ir(ABC)- Tr(BCA)= Ir(CAB (16) Tr(aa (17) 1.2 Determinant Let a be an nm× n matrix. (18) det(A),ifA∈R (19) det(a) det(a) (20) dct(AB) t(a)dct(B) (21) det(a) /det(A) det(a") det(A det(I+uvt) let(I+A)=1+det(A)+Tr(A (25) For nm-3 det(i+a)=1+det(a)+ Tr(a)+Tr(A Tr(A (26) peteRsen pedersen, THe Matrix CoOKBOOK, version: NoVEMBER 15, 2012, Page 6 1.3 The Special Case 2x2 1 BASICS F (+A + det(A)+Tr(a)+ +Tr(A Ir(A2) Tr(a Tr(A)Tr(A)+. Tr(A For small a, the following approximation holds +A)≌1-de(A)+m(A)+1=2(A)2-12I(A2)(28) 1.3 The Special Case 2x2 Consider the matrix a 41141 A214 Deterinillant and trace det(A)=A11422-A12A Eigenv Tr(A)+ det(a)=0 Tr(A)+VTr(A)2-4det(A 1 A,- Tr(A)vTr()2-4 det(A) A1+A2-1r(A 入1A2=det(A E A 入1-A Iverse A1 A214 (31) peteRsen pedersen, The Matrix CooKBooK, version: NoVEMBER 15, 2012, Page 2 DERIVATIVES 2 Derivatives This section is covering differentiation of a number of expressions with respect to a nlatrix X. Note that it is always assumed that X has TO special structure, i.e that the elements of X are independent(e.g. not symmetric, Toeplitz, positive dcfinitc). Scc scction 2.8 for diffcrcntiation of structured matrices. The basic assumptions can be written in a formula as aX that is for e. g. vector forms 1=(]= 0 The following rules are general and very useful when deriving the differential of an expresslon(①9) JA 0 (A is a coIlstant)(33) dax= aox 0(X+Y oX+OY (35 a(Tr(X))= Tr(aX a(XY (axY+X(OY) a(xc Y (aX)cY+Xo(ar (38 (x⑧Y (dX)⑧Y+X⑧(Y) (X-1 X-(aX)X a(dct(X))= Tr(adj(X)aX a(det(x)= det(x)Tr(x ax (42 a(n(det(X Tr(x aX) (43 (X)2 ax=(ax) (45) 2.1 Derivatives of a determinant 2.1.1 General form a dct(Y) aY 0, det(y)tr a: r a det(x aX Sir det(x) a2 det(Y) dY det(y) TrY O 0 Y Y Y peteRsen pedersen, THe Matrix CoOKBOOK, version: NoVEMBER 15, 2012, Page 8 2.2 Derivatives of an Inverse 2 DERIVATIVES 2.1.2 Linear forms a det (X) aX det(x)(x) ∑ a det(X) OX dij det(x) (50) a det(AxB) det(axB)(x= det(AXB)(X)(51) 2.1.3 Square forms If X is square anld invertible, theI a dct(X aX) 2 det(XAX)X-T (52) If X is not square but A is symmetric, then a det(xAx 2 det(x AXAX(x AX (53) aX If X is not square and A is not symmetric, then a det(xax) OX det(x AX)(AX(X AX)AX(Xa x))(54) 2.1.4 Other nonlinear forms Some special cases are(See 97) a In det(xx) aX =2(X (55) Oln det(x'X 2X aIn det (x) aX (X (X (57) adet(x k det(x)X (58) 2.2 Derivatives of an inverse From 27 we have the basic identity Y Y peteRsen pedersen, THe Matrix CoOKBOOK, version: NoVEMBER 15, 2012, Page 9 2.3 Derivatives of Eigenvalues 2 DERIVATIVES from which it follows (X-1)k OX (X-1)k(X-) dab X ab x 0ⅹ a det(x) det(x )(X ax OTr(AX B) (X BAX l、T ax (63) dTr((X+A)) ax (X+A)-1(X+A From 32 wc have the following result: Lct A bc an n x n invcrtiblc squarc matrix,w he the inverse of A, and (A) is an nx n-variate and differentiable function wit h respect to A, then the partia l differentia Is of with respect to a nd w satisfy a/ a A OA A 2. 3 Derivatives of Eigenvalues ∑eig(X) OX ig(X) det(X) det(x)x If A is real and syIllnetric. A i and vi are distinct eigenvalues and eigelvec tors of A (see(276)with vT Vi-1, then 33 O入 10(A) (AⅠ-A)+a(Avz 2.4 Derivatives of matrices vectors and scalar forms 2.4.1 First Order oxa x dx daB = ab x dX (71) da Xa daTtA (72) OX (73) a(XA)i X (J"A) a(xa OX Din(a (J"A)ij (75) petersen PEDERSEN, THE MATIIX COOKBOOK, VEnSION: NOVEMBER 15, 2012, Page 10 【实例截图】
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