The Matrix Cookbook. [ ] Kaare Brandt Petersen Michael Syskind Pedersen. Version: November 15, 2012


 Ashley O’Connor’
 2 years ago
 Views:
Transcription
1 The Matrix Cookbook ] Kaare Brandt Petersen Michael Syskind Pedersen Version: November 15, 01 1
2 Introduction What 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 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 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 Its ongoing: The project of keeping a large repository of relations involving matrices is naturally ongoing 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 Keywords: Matrix algebra, matrix relations, matrix identities, derivative of determinant, derivative of inverse matrix, differentiate a matrix. Acknowledgements: We would like to thank the following for contributions and suggestions: Bill Baxter, Brian Templeton, Christian Rishøj, Christian Schröppel, Dan Boley, Douglas L. Theobald, Esben HoeghRasmussen, Evripidis Karseras, Georg Martius, Glynne Casteel, Jan Larsen, Jun Bin Gao, Jürgen Struckmeier, Kamil Dedecius, Karim T. AbouMoustafa, Korbinian Strimmer, Lars Christiansen, Lars Kai Hansen, Leland Wilkinson, Liguo He, Loic Thibaut, Markus Froeb, Michael Hubatka, Miguel Barão, Ole Winther, Pavel Sakov, Stephan Hattinger, Troels Pedersen, Vasile 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, 01, Page
3 CONTENTS CONTENTS Contents 1 Basics Trace Determinant The Special Case x Derivatives 8.1 Derivatives of a Determinant Derivatives of an Inverse Derivatives of Eigenvalues Derivatives of Matrices, Vectors and Scalar Forms Derivatives of Traces Derivatives of vector norms Derivatives of matrix norms Derivatives of Structured Matrices Inverses Basic Exact Relations Implication on Inverses Approximations Generalized Inverse Pseudo Inverse Complex Matrices Complex Derivatives Higher order and nonlinear derivatives Inverse of complex sum Solutions and Decompositions Solutions to linear equations Eigenvalues and Eigenvectors Singular Value Decomposition Triangular Decomposition LU decomposition LDM decomposition LDL decompositions Statistics and Probability Definition of Moments Expectation of Linear Combinations Weighted Scalar Variable Multivariate Distributions Cauchy Dirichlet Normal NormalInverse Gamma Gaussian Multinomial Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 3
4 CONTENTS CONTENTS 7.7 Student s t Wishart Wishart, Inverse Gaussians Basics Moments Miscellaneous Mixture of Gaussians Special Matrices Block matrices Discrete Fourier Transform Matrix, The Hermitian Matrices and skewhermitian Idempotent Matrices Orthogonal matrices Positive Definite and Semidefinite Matrices Singleentry Matrix, The Symmetric, Skewsymmetric/Antisymmetric Toeplitz Matrices Transition matrices Units, Permutation and Shift Vandermonde Matrices Functions and Operators Functions and Series Kronecker and Vec Operator Vector Norms Matrix Norms Rank Integral Involving Dirac Delta Functions Miscellaneous A Onedimensional Results 64 A.1 Gaussian A. One Dimensional Mixture of Gaussians B Proofs and Details 66 B.1 Misc Proofs Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 4
5 CONTENTS CONTENTS Notation and Nomenclature A Matrix A ij Matrix indexed for some purpose A i Matrix indexed for some purpose A ij Matrix indexed for some purpose A n 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) A 1/ The square root of a matrix (if unique), not elementwise (A) ij The (i, j).th entry of the matrix A A ij The (i, j).th entry of the matrix A A] ij The ijsubmatrix, i.e. A with i.th row and j.th column deleted a Vector (columnvector) a i Vector indexed for some purpose a i The i.th element of the vector a a Scalar Rz Rz RZ Iz Iz IZ Real part of a scalar Real part of a vector Real part of a matrix Imaginary part of a scalar Imaginary part of a vector Imaginary part of a matrix det(a) Determinant of A Tr(A) Trace of the matrix A diag(a) Diagonal matrix of the matrix A, i.e. (diag(a)) ij = δ ij A ij eig(a) Eigenvalues of the matrix A vec(a) The vectorversion of the matrix A (see Sec. 10..) sup Supremum of a set A Matrix norm (subscript if any denotes what norm) A T Transposed matrix A T The inverse of the transposed and vice versa, A T = (A 1 ) T = (A T ) 1. A Complex conjugated matrix A H Transposed and complex conjugated matrix (Hermitian) A B A B Hadamard (elementwise) product Kronecker product 0 The null matrix. Zero in all entries. I The identity matrix The singleentry matrix, 1 at (i, j) and zero elsewhere Σ A positive definite matrix Λ A diagonal matrix J ij Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 5
6 1 BASICS 1 Basics (AB) 1 = B 1 A 1 (1) (ABC...) 1 =...C 1 B 1 A 1 () (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) 1.1 Trace Tr(A) = i A ii (11) Tr(A) = i λ i, λ i = eig(a) (1) 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) a T a = Tr(aa T ) (17) 1. Determinant Let A be an n n matrix. det(a) = i λ i λ i = eig(a) (18) det(ca) = c n det(a), if A R n n (19) det(a T ) = det(a) (0) det(ab) = det(a) det(b) (1) det(a 1 ) = 1/ det(a) () det(a n ) = det(a) n (3) det(i + uv T ) = 1 + u T v (4) For n = : For n = 3: det(i + A) = 1 + det(a) + Tr(A) (5) det(i + A) = 1 + det(a) + Tr(A) + 1 Tr(A) 1 Tr(A ) (6) Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 6
7 1.3 The Special Case x 1 BASICS For n = 4: det(i + A) = 1 + det(a) + Tr(A) + 1 +Tr(A) 1 Tr(A ) Tr(A)3 1 Tr(A)Tr(A ) Tr(A3 ) (7) For small ε, the following approximation holds det(i + εa) = 1 + det(a) + εtr(a) + 1 ε Tr(A) 1 ε Tr(A ) (8) 1.3 The Special Case x Consider the matrix A A11 A A = 1 A 1 A Determinant and trace ] Eigenvalues λ 1 = Tr(A) + Tr(A) 4 det(a) λ 1 + λ = Tr(A) Eigenvectors Inverse v 1 det(a) = A 11 A A 1 A 1 (9) Tr(A) = A 11 + A (30) λ λ Tr(A) + det(a) = 0 ] A 1 λ 1 A 11 A 1 = λ = Tr(A) Tr(A) 4 det(a) λ 1 λ = det(a) v A 1 λ A 11 det(a) A 1 A 11 1 A A 1 ] ] (31) Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 7
8 DERIVATIVES Derivatives This section 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.8 for differentiation of structured matrices. The basic assumptions can be written in a formula as X kl X ij = δ ik δ lj (3) that is for e.g. vector forms, ] x = x i y y i ] x = x y i y i ] x = x i y ij y j The following rules are general and very useful when deriving the differential of an expression (19]): A = 0 (A is a constant) (33) (αx) = αx (34) (X + Y) = X + Y (35) (Tr(X)) = Tr(X) (36) (XY) = (X)Y + X(Y) (37) (X Y) = (X) Y + X (Y) (38) (X Y) = (X) Y + X (Y) (39) (X 1 ) = X 1 (X)X 1 (40) (det(x)) = Tr(adj(X)X) (41) (det(x)) = det(x)tr(x 1 X) (4) (ln(det(x))) = Tr(X 1 X) (43) X T = (X) T (44) X H = (X) H (45).1 Derivatives of a Determinant.1.1 General form k det(y) x det(x) = det(y)tr Y 1 Y x ] (46) X jk = δ ij det(x) (47) X ik ] Y det(y) 1 x x = det(y) Tr Y x +Tr Y ( Tr Y 1 Y x 1 Y x ] Tr Y ) ( Y 1 Y ] x )] ] (48) 1 Y x Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 8
9 . Derivatives of an Inverse DERIVATIVES.1. Linear forms det(x) = det(x)(x 1 ) T (49) X det(x) X jk = δ ij det(x) (50) X ik k det(axb) X.1.3 Square forms If X is square and invertible, then = det(axb)(x 1 ) T = det(axb)(x T ) 1 (51) det(x T AX) X If X is not square but A is symmetric, then det(x T AX) X If X is not square and A is not symmetric, then det(x T AX) X.1.4 Other nonlinear forms Some special cases are (See 9, 7]) = det(x T AX)X T (5) = det(x T AX)AX(X T AX) 1 (53) = det(x T AX)(AX(X T AX) 1 + A T X(X T A T X) 1 ) (54) ln det(x T X) X = (X + ) T (55) ln det(x T X) X + = X T (56) ln det(x) X = (X 1 ) T = (X T ) 1 (57) det(x k ) X = k det(x k )X T (58). Derivatives of an Inverse From 7] we have the basic identity Y 1 x Y = Y 1 x Y 1 (59) Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 9
10 .3 Derivatives of Eigenvalues DERIVATIVES from which it follows (X 1 ) kl X ij = (X 1 ) ki (X 1 ) jl (60) a T X 1 b X = X T ab T X T (61) det(x 1 ) X = det(x 1 )(X 1 ) T (6) Tr(AX 1 B) X = (X 1 BAX 1 ) T (63) Tr((X + A) 1 ) X = ((X + A) 1 (X + A) 1 ) T (64) From 3] we have the following result: Let A be an n n invertible square matrix, W be the inverse of A, and J(A) is an n n variate and differentiable function with respect to A, then the partial differentials of J with respect to A and W satisfy J A = J A T W A T.3 Derivatives of Eigenvalues eig(x) = Tr(X) = I (65) X X eig(x) = X X det(x) = det(x)x T (66) If A is real and symmetric, λ i and v i are distinct eigenvalues and eigenvectors of A (see (76)) with vi T v i = 1, then 33] λ i = v T i (A)v i (67) v i = (λ i I A) + (A)v i (68).4 Derivatives of Matrices, Vectors and Scalar Forms.4.1 First Order x T a x a T Xb X a T X T b X a T Xa X = at x x = a (69) = ab T (70) = ba T (71) = at X T a X = aa T (7) X X ij = J ij (73) (XA) ij X mn = δ im (A) nj = (J mn A) ij (74) (X T A) ij X mn = δ in (A) mj = (J nm A) ij (75) Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 10
11 .4 Derivatives of Matrices, Vectors and Scalar Forms DERIVATIVES.4. Second Order X ij X kl (76) X kl X mn klmn = kl b T X T Xc X = X(bc T + cb T ) (77) (Bx + b) T C(Dx + d) x = B T C(Dx + d) + D T C T (Bx + b) (78) (X T BX) kl X ij = δ lj (X T B) ki + δ kj (BX) il (79) (X T BX) X ij = X T BJ ij + J ji BX (J ij ) kl = δ ik δ jl (80) See Sec 9.7 for useful properties of the Singleentry matrix J ij x T Bx x = (B + B T )x (81) b T X T DXc X = D T Xbc T + DXcb T (8) X (Xb + c)t D(Xb + c) = (D + D T )(Xb + c)b T (83) Assume W is symmetric, then s (x As)T W(x As) = A T W(x As) (84) x (x s)t W(x s) = W(x s) (85) s (x s)t W(x s) = W(x s) (86) x (x As)T W(x As) = W(x As) (87) A (x As)T W(x As) = W(x As)s T (88) As a case with complex values the following holds (a x H b) x This formula is also known from the LMS algorithm 14].4.3 Higherorder and nonlinear For proof of the above, see B.1.3. = b(a x H b) (89) (X n n 1 ) kl = (X r J ij X n 1 r ) kl (90) X ij r=0 n 1 X at X n b = (X r ) T ab T (X n 1 r ) T (91) r=0 Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 11
12 .5 Derivatives of Traces DERIVATIVES X at (X n ) T X n b = n 1 X n 1 r ab T (X n ) T X r r=0 +(X r ) T X n ab T (X n 1 r ) T ] (9) See B.1.3 for a proof. Assume s and r are functions of x, i.e. s = s(x), r = r(x), and that A is a constant, then x st Ar = ] T s Ar + x ] T r A T s (93) x (Ax) T (Ax) x (Bx) T (Bx) = x T A T Ax x x T B T Bx (94) = AT Ax x T BBx xt A T AxB T Bx (x T B T Bx) (95).4.4 Gradient and Hessian Using the above we have for the gradient and the Hessian.5 Derivatives of Traces f = x T Ax + b T x (96) x f = f x = (A + AT )x + b (97) f xx T = A + A T (98) Assume F (X) to be a differentiable function of each of the elements of X. It then holds that Tr(F (X)) = f(x) T X where f( ) is the scalar derivative of F ( )..5.1 First Order Tr(X) X = I (99) X Tr(XA) = AT (100) X Tr(AXB) = AT B T (101) X Tr(AXT B) = BA (10) X Tr(XT A) = A (103) X Tr(AXT ) = A (104) Tr(A X) X = Tr(A)I (105) Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 1
13 .5 Derivatives of Traces DERIVATIVES.5. Second Order See 7]. X Tr(X ) = X T (106) X Tr(X B) = (XB + BX) T (107) X Tr(XT BX) = BX + B T X (108) X Tr(BXXT ) = BX + B T X (109) X Tr(XXT B) = BX + B T X (110) X Tr(XBXT ) = XB T + XB (111) X Tr(BXT X) = XB T + XB (11) X Tr(XT XB) = XB T + XB (113) X Tr(AXBX) = AT X T B T + B T X T A T (114) X Tr(XT X) = X Tr(XXT ) = X (115) X Tr(BT X T CXB) = C T XBB T + CXBB T (116) X Tr X T BXC ] = BXC + B T XC T (117) X Tr(AXBXT C) = A T C T XB T + CAXB (118) ] X Tr (AXB + C)(AXB + C) T = A T (AXB + C)B T (119).5.3 Higher Order Tr(X X) = Tr(X)Tr(X) = Tr(X)I(10) X X X Tr(Xk ) = k(x k 1 ) T (11) X Tr(AXk ) = k 1 (X r AX k r 1 ) T (1) r=0 X Tr B T X T CXX T CXB ] = CXX T CXBB T +C T XBB T X T C T X +CXBB T X T CX +C T XX T C T XBB T (13) Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 13
14 .6 Derivatives of vector norms DERIVATIVES.5.4 Other X Tr(AX 1 B) = (X 1 BAX 1 ) T = X T A T B T X T (14) Assume B and C to be symmetric, then ] X Tr (X T CX) 1 A ] X Tr (X T CX) 1 (X T BX) ] X Tr (A + X T CX) 1 (X T BX) See 7]. = (CX(X T CX) 1 )(A + A T )(X T CX) 1 (15) = CX(X T CX) 1 X T BX(X T CX) 1 +BX(X T CX) 1 (16) = CX(A + X T CX) 1 X T BX(A + X T CX) 1 +BX(A + X T CX) 1 (17) Tr(sin(X)) X = cos(x) T (18).6 Derivatives of vector norms.6.1 Twonorm x x a = x a (19) x a x a I (x a)(x a)t = x x a x a x a 3.7 Derivatives of matrix norms For more on matrix norms, see Sec (130) x x = xt x = x (131) x.7.1 Frobenius norm X X F = X Tr(XXH ) = X (13) See (48). Note that this is also a special case of the result in equation Derivatives of Structured Matrices Assume 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 function f(a) df = ] ] T f A kl f A = Tr (133) da ij A kl A ij A A ij kl Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 14
15 .8 Derivatives of Structured Matrices DERIVATIVES The matrix differentiated with respect to itself is in this document referred to as the structure matrix of A and is defined simply by A A ij = S ij (134) If A has no special structure we have simply S ij = J ij, that is, the structure matrix is simply the singleentry matrix. Many structures have a representation in singleentry matrices, see Sec for more examples of structure matrices..8.1 The Chain Rule Sometimes the objective is to find the derivative of a matrix which is a function of another matrix. Let U = f(x), the goal is to find the derivative of the function g(u) with respect to X: g(u) X = g(f(x)) X Then the Chain Rule can then be written the following way: (135) g(u) X = g(u) x ij = M N k=1 l=1 Using matrix notation, this can be written as:.8. Symmetric g(u) X ij = Tr ( g(u) U g(u) u kl (136) u kl x ij )T U X ij ]. (137) If A is symmetric, then S ij = J ij + J ji J ij J ij and therefore ] ] T ] df f f f da = + diag A A A (138) That is, e.g., (5]): Tr(AX) X det(x) X ln det(x) X = A + A T (A I), see (14) (139) = det(x)(x 1 (X 1 I)) (140) = X 1 (X 1 I) (141).8.3 Diagonal If X is diagonal, then (19]): Tr(AX) X = A I (14) Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 15
16 .8 Derivatives of Structured Matrices DERIVATIVES.8.4 Toeplitz Like symmetric matrices and diagonal matrices also Toeplitz matrices has a special structure which should be taken into account when the derivative with respect to a matrix with Toeplitz structure. Tr(AT) T = Tr(TA) T = Tr(A) Tr(A T ] n1 ) Tr(A T ] 1n ] n 1, ) A n1 Tr(A T ] 1n )) Tr(A T ] 1n ],n 1 ) α(a) Tr(A) Tr(A T ]1n ] n 1, ) Tr(A T ]n1 ) A 1n Tr(A T ] 1n ],n 1 ) Tr(A T ] 1n )) Tr(A) (143) As it can be seen, the derivative α(a) also has a Toeplitz structure. Each value in the diagonal is the sum of all the diagonal valued in A, the values in the diagonals next to the main diagonal equal the sum of the diagonal next to the main diagonal in A T. This result is only valid for the unconstrained Toeplitz matrix. 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 (144) Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 16
17 3 INVERSES 3 Inverses 3.1 Basic Definition The inverse A 1 of a matrix A C n n is defined such that AA 1 = A 1 A = I, (145) 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. 1]) Cofactors and Adjoint The submatrix of a matrix A, denoted by A] ij is a (n 1) (n 1) matrix obtained by deleting the ith row and the jth column of A. The (i, j) cofactor of a matrix is defined as cof(a, i, j) = ( 1) i+j det(a] ij ), (146) 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) (147) The adjoint matrix is the transpose of the cofactor matrix adj(a) = (cof(a)) T, (148) Determinant The determinant of a matrix A C n n is defined as (see 1]) det(a) = = n ( 1) j+1 A 1j det (A] 1j ) (149) j=1 n A 1j cof(a, 1, j). (150) j= Construction The inverse matrix can be constructed, using the adjoint matrix, by A 1 = For the case of matrices, see section adj(a) (151) det(a) Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 17
18 3. Exact Relations 3 INVERSES Condition number The condition number of a matrix c(a) is the ratio between the largest and the smallest singular value of a matrix (see Section 5.3 on singular values), c(a) = d + d (15) The condition number can be used to measure how singular a matrix is. If the condition number is large, it indicates that the matrix is nearly singular. The condition number can also be estimated from the matrix norms. Here c(a) = A A 1, (153) where is a norm such as e.g the 1norm, the norm, the norm or the Frobenius norm (see Sec 10.4 for more on matrix norms). The norm of A equals (max(eig(a H A))) 1, p.57]. For a symmetric matrix, this reduces to A = max( eig(a) ) 1, p.394]. If the matrix is symmetric and positive definite, A = max(eig(a)). The condition number based on the norm thus reduces to A A 1 = max(eig(a)) max(eig(a 1 )) = max(eig(a)) min(eig(a)). (154) 3. Exact Relations 3..1 Basic 3.. The Woodbury identity (AB) 1 = B 1 A 1 (155) The Woodbury identity comes in many variants. The latter of the two can be found in 1] (A + CBC T ) 1 = A 1 A 1 C(B 1 + C T A 1 C) 1 C T A 1 (156) (A + UBV) 1 = A 1 A 1 U(B 1 + VA 1 U) 1 VA 1 (157) If P, R are positive definite, then (see 30]) (P 1 + B T R 1 B) 1 B T R 1 = PB T (BPB T + R) 1 (158) 3..3 The Kailath Variant See 4, page 153]. (A + BC) 1 = A 1 A 1 B(I + CA 1 B) 1 CA 1 (159) 3..4 ShermanMorrison (A + bc T ) 1 = A 1 A 1 bc T A c T A 1 b (160) Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 18
19 3. Exact Relations 3 INVERSES 3..5 The Searle Set of Identities The following set of identities, can be found in 5, page 151], (I + A 1 ) 1 = A(A + I) 1 (161) (A + BB T ) 1 B = A 1 B(I + B T A 1 B) 1 (16) (A 1 + B 1 ) 1 = A(A + B) 1 B = B(A + B) 1 A (163) A A(A + B) 1 A = B B(A + B) 1 B (164) A 1 + B 1 = A 1 (A + B)B 1 (165) (I + AB) 1 = I A(I + BA) 1 B (166) (I + AB) 1 A = A(I + BA) 1 (167) 3..6 Rank1 update of inverse of inner product Denote A = (X T X) 1 and that X is extended to include a new column vector in the end X = X v]. Then 34] ] ( X T 1 A + X) AXT vv T XA T AX T v = v T v v T XAX T v v T v v T XAX T v v T XA T v T v v T XAX T v 3..7 Rank1 update of MoorePenrose Inverse 1 v T v v T XAX T v The following is a rank1 update for the MoorePenrose pseudoinverse of real valued matrices and proof can be found in 18]. The matrix G is defined below: Using the the notation (A + cd T ) + = A + + G (168) β = 1 + d T A + c (169) v = A + c (170) n = (A + ) T d (171) w = (I AA + )c (17) m = (I A + A) T d (173) the solution is given as six different cases, depending on the entities w, m, and β. Please note, that for any (column) vector v it holds that v + = v T (v T v) 1 = vt v. The solution is: Case 1 of 6: If w 0 and m 0. Then G = vw + (m + ) T n T + β(m + ) T w + (174) = 1 w vwt 1 β m mnt + m w mwt (175) Case of 6: If w = 0 and m 0 and β = 0. Then G = vv + A + (m + ) T n T (176) = 1 v vvt A + 1 m mnt (177) Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 19
20 3.3 Implication on Inverses 3 INVERSES Case 3 of 6: If w = 0 and β 0. Then G = 1 ( ) ( ) β mvt A + β v m T v m + β β m + v β (A+ ) T v + n (178) Case 4 of 6: If w 0 and m = 0 and β = 0. Then G = A + nn + vw + (179) = 1 n A+ nn T 1 w vwt (180) Case 5 of 6: If m = 0 and β 0. Then G = 1 β A+ nw T β n w + β ( w Case 6 of 6: If w = 0 and m = 0 and β = 0. Then β ) ( ) n T A+ n + v β w + n (181) G = vv + A + A + nn + + v + A + nvn + (18) = 1 v vvt A + 1 n A+ nn T + vt A + n v n vnt (183) 3.3 Implication on Inverses See 5]. If (A + B) 1 = A 1 + B 1 then AB 1 A = BA 1 B (184) A PosDef identity Assume P, R to be positive definite and invertible, then See 30]. (P 1 + B T R 1 B) 1 B T R 1 = PB T (BPB T + R) 1 (185) 3.4 Approximations The following identity is known as the Neuman series of a matrix, which holds when λ i < 1 for all eigenvalues λ i which is equivalent to (I A) 1 = (I + A) 1 = A n (186) n=0 ( 1) n A n (187) When λ i < 1 for all eigenvalues λ i, it holds that A 0 for n, and the following approximations holds n=0 (I A) 1 = I + A + A (I + A) 1 = I A + A (188) (189) Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 0
21 3.5 Generalized Inverse 3 INVERSES The following approximation is from ] and holds when A large and symmetric If σ is small compared to Q and M then Proof: A A(I + A) 1 A = I A 1 (190) (Q + σ M) 1 = Q 1 σ Q 1 MQ 1 (191) (Q + σ M) 1 = (19) (QQ 1 Q + σ MQ 1 Q) 1 = (193) This can be rewritten using the Taylor expansion: ((I + σ MQ 1 )Q) 1 = (194) Q 1 (I + σ MQ 1 ) 1 (195) Q 1 (I + σ MQ 1 ) 1 = (196) Q 1 (I σ MQ 1 + (σ MQ 1 )...) = Q 1 σ Q 1 MQ 1 (197) 3.5 Generalized Inverse Definition A generalized inverse matrix of the matrix A is any matrix A such that (see 6]) AA A = A (198) The matrix A is not unique. 3.6 Pseudo Inverse Definition The pseudo inverse (or MoorePenrose inverse) of a matrix A is the matrix A + that fulfils I AA + A = A II A + AA + = A + III IV AA + symmetric A + A symmetric The matrix A + is unique and does always exist. Note that in case of complex matrices, the symmetric condition is substituted by a condition of being Hermitian. Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 1
22 3.6 Pseudo Inverse 3 INVERSES 3.6. Properties Assume A + to be the pseudoinverse of A, then (See 3] for some of them) (A + ) + = A (199) (A T ) + = (A + ) T (00) (A H ) + = (A + ) H (01) (A ) + = (A + ) (0) (A + A)A H = A H (03) (A + A)A T A T (04) (ca) + = (1/c)A + (05) A + = (A T A) + A T (06) A + = A T (AA T ) + (07) (A T A) + = A + (A T ) + (08) (AA T ) + = (A T ) + A + (09) A + = (A H A) + A H (10) A + = A H (AA H ) + (11) (A H A) + = A + (A H ) + (1) (AA H ) + = (A H ) + A + (13) (AB) + = (A + AB) + (ABB + ) + (14) f(a H A) f(0)i = A + f(aa H ) f(0)i]a (15) f(aa H ) f(0)i = Af(A H A) f(0)i]a + (16) where A C n m. Assume A to have full rank, then For two matrices it hold that Construction (AA + )(AA + ) = AA + (17) (A + A)(A + A) = A + A (18) Assume that A has full rank, then Tr(AA + ) = rank(aa + ) (See 6]) (19) Tr(A + A) = rank(a + A) (See 6]) (0) (AB) + = (A + AB) + (ABB + ) + (1) (A B) + = A + B + () A n n Square rank(a) = n A + = A 1 A n m Broad rank(a) = n A + = A T (AA T ) 1 A n m Tall rank(a) = m A + = (A T A) 1 A T The socalled broad version is also known as right inverse and the tall version as the left inverse. Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page
23 3.6 Pseudo Inverse 3 INVERSES Assume 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 = UDV T, by A + = V r D 1 r U T r (3) where U r, D r, and V r are the matrices with the degenerated rows and columns deleted. A different way is this: There do always exist two matrices C n r and D r m of rank r, such that A = CD. Using these matrices it holds that See 3]. A + = D T (DD T ) 1 (C T C) 1 C T (4) Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 3
24 4 COMPLEX MATRICES 4 Complex Matrices The complex scalar product r = pq can be written as ] ] ] Rr Rp Ip Rq = Ir Ip Rp Iq (5) 4.1 Complex Derivatives In order to differentiate an expression f(z) with respect to a complex z, the CauchyRiemann equations have to be satisfied (7]): df(z) dz and df(z) dz or in a more compact form: = R(f(z)) Rz = i R(f(z)) Iz f(z) Iz + i I(f(z)) Rz + I(f(z)) Iz (6) (7) = if(z) Rz. (8) A complex function that satisfies the CauchyRiemann equations for points in a region R is said yo be analytic in this region R. In general, expressions involving complex conjugate or conjugate transpose do not satisfy the CauchyRiemann equations. In order to avoid this problem, a more generalized definition of complex derivative is used (4], 6]): Generalized Complex Derivative: df(z) dz = 1 Conjugate Complex Derivative ( f(z) ) Rz if(z). (9) Iz df(z) dz = 1 ( f(z) ) Rz + if(z). (30) Iz The Generalized Complex Derivative equals the normal derivative, when f is an analytic function. For a nonanalytic function such as f(z) = z, the derivative equals zero. The Conjugate Complex Derivative equals zero, when f is an analytic function. The Conjugate Complex Derivative has e.g been used by 1] when deriving a complex gradient. Notice: df(z) dz f(z) Rz + if(z) Iz. (31) Complex Gradient Vector: If f is a real function of a complex vector z, then the complex gradient vector is given by (14, p. 798]) f(z) = df(z) dz (3) = f(z) Rz + if(z) Iz. Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 4
25 4.1 Complex Derivatives 4 COMPLEX MATRICES Complex Gradient Matrix: If f is a real function of a complex matrix Z, then the complex gradient matrix is given by (]) f(z) = df(z) dz (33) = f(z) RZ + if(z) IZ. These expressions can be used for gradient descent algorithms The Chain Rule for complex numbers The chain rule is a little more complicated when the function of a complex u = f(x) is nonanalytic. For a nonanalytic function, the following chain rule can be applied (7]) g(u) x = g u u x + g u u x = g u ( g ) u u x + u x (34) Notice, if the function is analytic, the second term reduces to zero, and the function is reduced to the normal wellknown chain rule. For the matrix derivative of a scalar function g(u), the chain rule can be written the following way: g(u) X g(u) Tr(( = U )T U) + X 4.1. Complex Derivatives of Traces Tr(( g(u) U ) T U ) X. (35) If the derivatives involve complex numbers, the conjugate transpose is often involved. The most useful way to show complex derivative is to show the derivative with respect to the real and the imaginary part separately. An easy example is: Tr(X ) RX = Tr(XH ) RX i Tr(X ) IX = ) itr(xh IX = I (36) = I (37) Since the two results have the same sign, the conjugate complex derivative (30) should be used. Tr(X) RX = Tr(XT ) RX i Tr(X) IX = ) itr(xt IX = I (38) = I (39) Here, the two results have different signs, and the generalized complex derivative (9) should be used. Hereby, it can be seen that (100) holds even if X is a complex number. Tr(AX H ) RX i Tr(AXH ) IX = A (40) = A (41) Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 5
26 4. Higher order and nonlinear derivatives 4 COMPLEX MATRICES Tr(AX ) RX i Tr(AX ) IX = A T (4) = A T (43) Tr(XX H ) RX i Tr(XXH ) IX = Tr(XH X) RX = i Tr(XH X) IX = RX (44) = iix (45) By inserting (44) and (45) in (9) and (30), it can be seen that Tr(XX H ) = X X (46) Tr(XX H ) X = X (47) Since the function Tr(XX H ) is a real function of the complex matrix X, the complex gradient matrix (33) is given by Tr(XX H ) = Tr(XXH ) X = X (48) Complex Derivative Involving Determinants Here, a calculation example is provided. The objective is to find the derivative of det(x H AX) with respect to X C m n. The derivative is found with respect to the real part and the imaginary part of X, by use of (4) and (37), det(x H AX) can be calculated as (see App. B.1.4 for details) det(x H AX) X = 1 ( det(x H AX) i det(xh AX) ) RX IX = det(x H AX) ( (X H AX) 1 X H A ) T (49) and the complex conjugate derivative yields det(x H AX) X = 1 ( det(x H AX) + i det(xh AX) ) RX IX = det(x H AX)AX(X H AX) 1 (50) 4. Higher order and nonlinear derivatives (Ax) H (Ax) x (Bx) H (Bx) = x H A H Ax x x H B H Bx (51) = AH Ax x H BBx xh A H AxB H Bx (x H B H Bx) (5) Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 01, Page 6
MATH36001 Background Material 2015
MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be
More informationThe Matrix Cookbook. Kaare Brandt Petersen Michael Syskind Pedersen. Version: January 5, 2005
The Matrix Cookbook Kaare Brandt Petersen Michael Syskind Pedersen Version: January 5, 2005 What is this? These pages are a collection of facts (identities, approximations, inequalities, relations,...)
More informationThe Matrix Cookbook. [ ] Kaare Brandt Petersen Michael Syskind Pedersen. Version: September 5, 2007
The Matrix Cookbook http://matrixcookbook.com ] Kaare Brandt Petersen Michael Syskind Pedersen Version: September 5, 007 What is this? These pages are a collection of facts (identities, approximations,
More informationR mp nq. (13.1) a m1 B a mn B
This electronic version is for personal use and may not be duplicated or distributed Chapter 13 Kronecker Products 131 Definition and Examples Definition 131 Let A R m n, B R p q Then the Kronecker product
More informationMathematics Notes for Class 12 chapter 3. Matrices
1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationApplied Linear Algebra I Review page 1
Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties
More informationMATH 240 Fall, Chapter 1: Linear Equations and Matrices
MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS
More information4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns
L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows
More information1 Introduction. 2 Matrices: Definition. Matrix Algebra. Hervé Abdi Lynne J. Williams
In Neil Salkind (Ed.), Encyclopedia of Research Design. Thousand Oaks, CA: Sage. 00 Matrix Algebra Hervé Abdi Lynne J. Williams Introduction Sylvester developed the modern concept of matrices in the 9th
More informationDeterminants. Dr. Doreen De Leon Math 152, Fall 2015
Determinants Dr. Doreen De Leon Math 52, Fall 205 Determinant of a Matrix Elementary Matrices We will first discuss matrices that can be used to produce an elementary row operation on a given matrix A.
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationEC9A0: Presessional Advanced Mathematics Course
University of Warwick, EC9A0: Presessional Advanced Mathematics Course Peter J. Hammond & Pablo F. Beker 1 of 55 EC9A0: Presessional Advanced Mathematics Course Slides 1: Matrix Algebra Peter J. Hammond
More informationequations Karl Lundengård December 3, 2012 MAA704: Matrix functions and matrix equations Matrix functions Matrix equations Matrix equations, cont d
and and Karl Lundengård December 3, 2012 Solving General, Contents of todays lecture and (Kroenecker product) Solving General, Some useful from calculus and : f (x) = x n, x C, n Z + : f (x) = n x, x R,
More informationCONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation
Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More informationADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB. Sohail A. Dianat. Rochester Institute of Technology, New York, U.S.A. Eli S.
ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB Sohail A. Dianat Rochester Institute of Technology, New York, U.S.A. Eli S. Saber Rochester Institute of Technology, New York, U.S.A. (g) CRC Press Taylor
More informationTopic 1: Matrices and Systems of Linear Equations.
Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method
More informationInverses. Stephen Boyd. EE103 Stanford University. October 27, 2015
Inverses Stephen Boyd EE103 Stanford University October 27, 2015 Outline Left and right inverses Inverse Solving linear equations Examples Pseudoinverse Left and right inverses 2 Left inverses a number
More informationLINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LUdecomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
More informationSolutions to Review Problems
Chapter 1 Solutions to Review Problems Chapter 1 Exercise 42 Which of the following equations are not linear and why: (a x 2 1 + 3x 2 2x 3 = 5. (b x 1 + x 1 x 2 + 2x 3 = 1. (c x 1 + 2 x 2 + x 3 = 5. (a
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationAdvanced Techniques for Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz
Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Vectors Arrays of numbers Vectors represent a point in a n dimensional
More informationSummary of week 8 (Lectures 22, 23 and 24)
WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry
More informationChapter 8. Matrices II: inverses. 8.1 What is an inverse?
Chapter 8 Matrices II: inverses We have learnt how to add subtract and multiply matrices but we have not defined division. The reason is that in general it cannot always be defined. In this chapter, we
More information3. Let A and B be two n n orthogonal matrices. Then prove that AB and BA are both orthogonal matrices. Prove a similar result for unitary matrices.
Exercise 1 1. Let A be an n n orthogonal matrix. Then prove that (a) the rows of A form an orthonormal basis of R n. (b) the columns of A form an orthonormal basis of R n. (c) for any two vectors x,y R
More informationUsing the Singular Value Decomposition
Using the Singular Value Decomposition Emmett J. Ientilucci Chester F. Carlson Center for Imaging Science Rochester Institute of Technology emmett@cis.rit.edu May 9, 003 Abstract This report introduces
More informationLinear Algebra: Determinants, Inverses, Rank
D Linear Algebra: Determinants, Inverses, Rank D 1 Appendix D: LINEAR ALGEBRA: DETERMINANTS, INVERSES, RANK TABLE OF CONTENTS Page D.1. Introduction D 3 D.2. Determinants D 3 D.2.1. Some Properties of
More informationMatrix Algebra, Class Notes (part 1) by Hrishikesh D. Vinod Copyright 1998 by Prof. H. D. Vinod, Fordham University, New York. All rights reserved.
Matrix Algebra, Class Notes (part 1) by Hrishikesh D. Vinod Copyright 1998 by Prof. H. D. Vinod, Fordham University, New York. All rights reserved. 1 Sum, Product and Transpose of Matrices. If a ij with
More information1 Determinants. Definition 1
Determinants The determinant of a square matrix is a value in R assigned to the matrix, it characterizes matrices which are invertible (det 0) and is related to the volume of a parallelpiped described
More informationPartitioned Matrices and the Schur Complement
P Partitioned Matrices and the Schur Complement P 1 Appendix P: PARTITIONED MATRICES AND THE SCHUR COMPLEMENT TABLE OF CONTENTS Page P1 Partitioned Matrix P 3 P2 Schur Complements P 3 P3 Block Diagonalization
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra  1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More informationFactorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 3 Linear Least Squares Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign Copyright c 2002. Reproduction
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More information(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.
Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product
More information2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors
2.1: Determinants by Cofactor Expansion Math 214 Chapter 2 Notes and Homework Determinants The minor M ij of the entry a ij is the determinant of the submatrix obtained from deleting the i th row and the
More informationMath 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.
Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(
More informationMatrix Inverse and Determinants
DM554 Linear and Integer Programming Lecture 5 and Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1 2 3 4 and Cramer s rule 2 Outline 1 2 3 4 and
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationSergei Silvestrov, Christopher Engström, Karl Lundengård, Johan Richter, Jonas Österberg. November 13, 2014
Sergei Silvestrov,, Karl Lundengård, Johan Richter, Jonas Österberg November 13, 2014 Analysis Todays lecture: Course overview. Repetition of matrices elementary operations. Repetition of solvability of
More informationFacts About Eigenvalues
Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v
More information13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.
3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in threespace, we write a vector in terms
More information5.3 Determinants and Cramer s Rule
290 5.3 Determinants and Cramer s Rule Unique Solution of a 2 2 System The 2 2 system (1) ax + by = e, cx + dy = f, has a unique solution provided = ad bc is nonzero, in which case the solution is given
More informationCalculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants
Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants Hartmut Führ fuehr@matha.rwthaachen.de Lehrstuhl A für Mathematik, RWTH Aachen October 30, 2008 Overview
More informationINTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL
SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics
More informationThe Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression
The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonaldiagonalorthogonal type matrix decompositions Every
More informationEconomics 102C: Advanced Topics in Econometrics 2  Tooling Up: The Basics
Economics 102C: Advanced Topics in Econometrics 2  Tooling Up: The Basics Michael Best Spring 2015 Outline The Evaluation Problem Matrix Algebra Matrix Calculus OLS With Matrices The Evaluation Problem:
More informationNotes on Symmetric Matrices
CPSC 536N: Randomized Algorithms 201112 Term 2 Notes on Symmetric Matrices Prof. Nick Harvey University of British Columbia 1 Symmetric Matrices We review some basic results concerning symmetric matrices.
More informationMATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.
MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An mbyn matrix is a rectangular array of numbers that has m rows and n columns: a 11
More information9. Numerical linear algebra background
Convex Optimization Boyd & Vandenberghe 9. Numerical linear algebra background matrix structure and algorithm complexity solving linear equations with factored matrices LU, Cholesky, LDL T factorization
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationSF2940: Probability theory Lecture 8: Multivariate Normal Distribution
SF2940: Probability theory Lecture 8: Multivariate Normal Distribution Timo Koski 24.09.2015 Timo Koski Matematisk statistik 24.09.2015 1 / 1 Learning outcomes Random vectors, mean vector, covariance matrix,
More information4. MATRICES Matrices
4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:
More informationMatrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.
Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that
More informationChapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6
Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a
More information1.5 Elementary Matrices and a Method for Finding the Inverse
.5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:
More informationNotes for STA 437/1005 Methods for Multivariate Data
Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.
More informationLecture 2 Matrix Operations
Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrixvector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or
More informationIntroduction to Matrix Algebra I
Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model
More informationAPPENDIX QA MATRIX ALGEBRA. a 11 a 12 a 1K A = [a ik ] = [A] ik = 21 a 22 a 2K. a n1 a n2 a nk
Greene2140242 book December 1, 2010 8:8 APPENDIX QA MATRIX ALGEBRA A.1 TERMINOLOGY A matrix is a rectangular array of numbers, denoted a 11 a 12 a 1K A = [a ik ] = [A] ik = a 21 a 22 a 2K. a n1 a n2 a
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationQuadratic Polynomials
Math 210 Quadratic Polynomials Jerry L. Kazdan Polynomials in One Variable. After studying linear functions y = ax + b, the next step is to study quadratic polynomials, y = ax 2 + bx + c, whose graphs
More information( % . This matrix consists of $ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4&
Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important
More informationNotes on Matrix Calculus
Notes on Matrix Calculus Paul L. Fackler North Carolina State University September 27, 2005 Matrix calculus is concerned with rules for operating on functions of matrices. For example, suppose that an
More informationNOTES on LINEAR ALGEBRA 1
School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura
More informationDeterminants. Chapter Properties of the Determinant
Chapter 4 Determinants Chapter 3 entailed a discussion of linear transformations and how to identify them with matrices. When we study a particular linear transformation we would like its matrix representation
More information10. Graph Matrices Incidence Matrix
10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationLinear Least Squares
Linear Least Squares Suppose we are given a set of data points {(x i,f i )}, i = 1,...,n. These could be measurements from an experiment or obtained simply by evaluating a function at some points. One
More informationEigenvalues, Eigenvectors, Matrix Factoring, and Principal Components
Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they
More information9 MATRICES AND TRANSFORMATIONS
9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the
More informationMatrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo
Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for
More informationEC327: Advanced Econometrics, Spring 2007
EC327: Advanced Econometrics, Spring 2007 Wooldridge, Introductory Econometrics (3rd ed, 2006) Appendix D: Summary of matrix algebra Basic definitions A matrix is a rectangular array of numbers, with m
More informationThe Solution of Linear Simultaneous Equations
Appendix A The Solution of Linear Simultaneous Equations Circuit analysis frequently involves the solution of linear simultaneous equations. Our purpose here is to review the use of determinants to solve
More informationCHARACTERISTIC ROOTS AND VECTORS
CHARACTERISTIC ROOTS AND VECTORS 1 DEFINITION OF CHARACTERISTIC ROOTS AND VECTORS 11 Statement of the characteristic root problem Find values of a scalar λ for which there exist vectors x 0 satisfying
More informationEigenvalues and Eigenvectors
Chapter 6 Eigenvalues and Eigenvectors 6. Introduction to Eigenvalues Linear equations Ax D b come from steady state problems. Eigenvalues have their greatest importance in dynamic problems. The solution
More informationLecture Notes 1: Matrix Algebra Part B: Determinants and Inverses
University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 57 Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Hammond email: p.j.hammond@warwick.ac.uk Autumn 2012,
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationLinear Algebra: Matrices
B Linear Algebra: Matrices B 1 Appendix B: LINEAR ALGEBRA: MATRICES TABLE OF CONTENTS Page B.1. Matrices B 3 B.1.1. Concept................... B 3 B.1.2. Real and Complex Matrices............ B 3 B.1.3.
More informationBasics Inversion and related concepts Random vectors Matrix calculus. Matrix algebra. Patrick Breheny. January 20
Matrix algebra January 20 Introduction Basics The mathematics of multiple regression revolves around ordering and keeping track of large arrays of numbers and solving systems of equations The mathematical
More informationSection 6.1  Inner Products and Norms
Section 6.1  Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
More informationMATRICES WITH DISPLACEMENT STRUCTURE A SURVEY
MATRICES WITH DISPLACEMENT STRUCTURE A SURVEY PLAMEN KOEV Abstract In the following survey we look at structured matrices with what is referred to as low displacement rank Matrices like Cauchy Vandermonde
More informationMatrix Algebra. Definitions and Operations
Chapter 1 Matrix Algebra. Definitions and Operations 1.1 Matrices Matrices play very important roles in the computation and analysis of several engineering problems. First, matrices allow for compact notations.
More informationMatrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,
LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x
More information9 Matrices, determinants, inverse matrix, Cramer s Rule
AAC  Business Mathematics I Lecture #9, December 15, 2007 Katarína Kálovcová 9 Matrices, determinants, inverse matrix, Cramer s Rule Basic properties of matrices: Example: Addition properties: Associative:
More informationSF2940: Probability theory Lecture 8: Multivariate Normal Distribution
SF2940: Probability theory Lecture 8: Multivariate Normal Distribution Timo Koski 24.09.2014 Timo Koski () Mathematisk statistik 24.09.2014 1 / 75 Learning outcomes Random vectors, mean vector, covariance
More informationLecture 11. Shuanglin Shao. October 2nd and 7th, 2013
Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous
More informationNumerical Linear Algebra Chap. 4: Perturbation and Regularisation
Numerical Linear Algebra Chap. 4: Perturbation and Regularisation Heinrich Voss voss@tuharburg.de Hamburg University of Technology Institute of Numerical Simulation TUHH Heinrich Voss Numerical Linear
More informationLecture 4: Partitioned Matrices and Determinants
Lecture 4: Partitioned Matrices and Determinants 1 Elementary row operations Recall the elementary operations on the rows of a matrix, equivalent to premultiplying by an elementary matrix E: (1) multiplying
More informationMAT188H1S Lec0101 Burbulla
Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u
More informationHypothesis Testing in the Classical Regression Model
LECTURE 5 Hypothesis Testing in the Classical Regression Model The Normal Distribution and the Sampling Distributions It is often appropriate to assume that the elements of the disturbance vector ε within
More informationCofactor Expansion: Cramer s Rule
Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating
More informationMATH 511 ADVANCED LINEAR ALGEBRA SPRING 2006
MATH 511 ADVANCED LINEAR ALGEBRA SPRING 26 Sherod Eubanks HOMEWORK 1 1.1 : 3, 5 1.2 : 4 1.3 : 4, 6, 12, 13, 16 1.4 : 1, 5, 8 Section 1.1: The EigenvalueEigenvector Equation Problem 3 Let A M n (R). If
More informationDETERMINANTS. b 2. x 2
DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in
More informationMatrices provide a compact notation for expressing systems of equations or variables. For instance, a linear function might be written as: b k
MATRIX ALGEBRA FOR STATISTICS: PART 1 Matrices provide a compact notation for expressing systems of equations or variables. For instance, a linear function might be written as: y = x 1 b 1 + x 2 + x 3
More information3.1 Least squares in matrix form
118 3 Multiple Regression 3.1 Least squares in matrix form E Uses Appendix A.2 A.4, A.6, A.7. 3.1.1 Introduction More than one explanatory variable In the foregoing chapter we considered the simple regression
More informationDerivatives of Matrix Functions and their Norms.
Derivatives of Matrix Functions and their Norms. Priyanka Grover Indian Statistical Institute, Delhi India February 24, 2011 1 / 41 Notations H : ndimensional complex Hilbert space. L (H) : the space
More information