The Matrix Cookbook. Kaare Brandt Petersen Michael Syskind Pedersen. Version: January 5, 2005

Transcription

1 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,...) 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 kbp@imm.dtu.dk. 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 at kbp@imm.dtu.dk. Acknowledgements: We would like to thank the following for discussions, proofreading, extensive corrections and suggestions: Esben Hoegh-Rasmussen and Vasile Sima. Keywords: Matrix algebra, matrix relations, matrix identities, derivative of determinant, derivative of inverse matrix, differentiate a matrix.

2 CONTENTS CONTENTS Contents Basics 5. Trace and Determinants The Special Case 2x Derivatives 7 2. Derivatives of a Determinant Derivatives of an Inverse Derivatives of Matrices, Vectors and Scalar Forms Derivatives of Traces Derivatives of Structured Matrices Inverses 4 3. Exact Relations Implication on Inverses Approximations Generalized Inverse Pseudo Inverse Complex Matrices 7 4. Complex Derivatives Decompositions Eigenvalues and Eigenvectors Singular Value Decomposition Triangular Decomposition General Statistics and Probability Moments of any distribution Expectations Gaussians One Dimensional Basics Moments Miscellaneous One Dimensional Mixture of Gaussians Mixture of Gaussians Miscellaneous 3 8. Functions and Series Indices, Entries and Vectors Solutions to Systems of Equations Block matrices Matrix Norms Positive Definite and Semi-definite Matrices Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 2

3 CONTENTS CONTENTS 8.7 Integral Involving Dirac Delta Functions Miscellaneous A Proofs and Details 4 Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 3

4 CONTENTS CONTENTS Notation and Nomenclature A A ij A i A ij A n A A + A /2 (A) ij A ij a a i a i a Matrix Matrix 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 matrix of the matrix A The pseudo inverse matrix of the matrix A The square root of a matrix (if unique), not elementwise The (i, j).th entry of the matrix A The (i, j).th entry of the matrix A Vector Vector indexed for some purpose The i.th element of the vector a Scalar Rz Rz RZ Iz Iz IZ det(a) A A T A A H A B A B 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 Determinant of A Matrix norm (subscript if any denotes what norm) Transposed matrix Complex conjugated matrix Transposed and complex conjugated matrix Hadamard (elementwise) product Kronecker product 0 The null matrix. Zero in all entries. I The identity matrix The single-entry matrix, at (i, j) and zero elsewhere Σ A positive definite matrix Λ A diagonal matrix J ij Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 4

5 BASICS Basics (AB) = B A (ABC...) =...C B A (A T ) = (A ) T (A + B) T = A T + B T (AB) T = B T A T (ABC...) T =...C T B T A T (A H ) = (A ) H (A + B) H = A H + B H (AB) H = B H A H (ABC...) H =...C H B H A H. Trace and Determinants Tr(A) = i A ii = i λ i, λ i = eig(a) Tr(A) = Tr(A T ) Tr(AB) = Tr(BA) Tr(A + B) = Tr(A) + Tr(B) Tr(ABC) = Tr(BCA) = Tr(CAB) det(a) = i λ i λ i = eig(a) det(ab) = det(a) det(b), det(a ) = if A and B are invertible det(a) Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 5

6 .2 The Special Case 2x2 BASICS.2 The Special Case 2x2 Consider the matrix A A A A = 2 A 2 A 22 Determinant and trace Eigenvalues λ = Tr(A) + Tr(A) 2 4 det(a) 2 λ + λ 2 = Tr(A) Eigenvectors Inverse v det(a) = A A 22 A 2 A 2 Tr(A) = A + A 22 λ 2 λ Tr(A) + det(a) = 0 A 2 λ A A = λ 2 = Tr(A) Tr(A) 2 4 det(a) 2 λ λ 2 = det(a) v 2 A22 A 2 det(a) A 2 A A 2 λ 2 A Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 6

7 2 DERIVATIVES 2 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 2.5 for differentiation of structured matrices. The basic assumptions can be written in a formula as X kl X ij = δ ik δ lj 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 (0): A = 0 (A is a constant) () (αx) = αx (2) (X + Y) = X + Y (3) (Tr(X)) = Tr(X) (4) (XY) = (X)Y + X(Y) (5) (X Y) = (X) Y + X (Y) (6) (X Y) = (X) Y + X (Y) (7) (X ) = X (X)X (8) (det(x)) = det(x)tr(x X) (9) (ln(det(x))) = Tr(X X) (0) X T = (X) T () X H = (X) H (2) 2. Derivatives of a Determinant 2.. General form 2..2 Linear forms det(axb) X det(y) x det(x) X = det(y)tr Y Y = det(x)(x ) T x = det(axb)(x ) T = det(axb)(x T ) Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 7

8 2.2 Derivatives of an Inverse 2 DERIVATIVES 2..3 Square forms If X is square and invertible, then det(x T AX) X If X is not square but A is symmetric, then det(x T AX) X = 2 det(x T AX)X T = 2 det(x T AX)AX(X T AX) If X is not square and A is not symmetric, then det(x T AX) X 2..4 Other nonlinear forms Some special cases are (See 8) = det(x T AX)(AX(X T AX) + A T X(X T A T X) ) (3) See 7. ln det(x T X) X = 2(X + ) T ln det(x T X) X + = 2X T ln det(x) = (X ) T = (X T ) X det(x k ) = k det(x k )X T X 2.2 Derivatives of an Inverse From 5 we have the basic identity from which it follows Y x Y = Y x Y (X ) kl X ij = (X ) ki (X ) jl a T X b = X T ab T X T X det(x ) = det(x )(X ) T X Tr(AX B) = (X BAX ) T X Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 8

9 2.3 Derivatives of Matrices, Vectors and Scalar Forms 2 DERIVATIVES 2.3 Derivatives of Matrices, Vectors and Scalar Forms 2.3. First Order x T b x a T Xa X a T Xb X a T X T b X = bt x x = abt = bat = b = at X T a X = aat X = J ij X ij (XA) ij X mn (X T A) ij X mn = δ im (A) nj = (J mn A) ij = δ in (A) mj = (J nm A) ij Second Order X ij klmn X kl X mn = 2 kl b T X T Xc X (Bx + b) T C(Dx + d) x (X T BX) kl X ij X kl = X(bc T + cb T ) = B T C(Dx + d) + D T C T (Bx + b) = δ lj (X T B) ki + δ kj (BX) il (X T BX) X ij = X T BJ ij + J ji BX (J ij ) kl = δ ik δ jl See Sec 8.2 for useful properties of the Single-entry matrix J ij x T Bx x = (B + B T )x b T X T DXc = D T Xbc T + DXcb T X X (Xb + c)t D(Xb + c) = (D + D T )(Xb + c)b T Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 9

10 2.3 Derivatives of Matrices, Vectors and Scalar Forms 2 DERIVATIVES Assume W is symmetric, then s (x As)T W(x As) = 2A T W(x As) x (x As)T W(x As) = 2W(x As) A (x As)T W(x As) = 2W(x As)s T Higher order and non-linear n X at X n b = (X r ) T ab T (X n r ) T (4) r=0 X at (X n ) T X n b = n X n r ab T (X n ) T X r r=0 +(X r ) T X n ab T (X n r ) T (5) See A.0. 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 T s x st As = (A + A T )s x T T s r x st Ar = As + A T r x x Gradient and Hessian Using the above we have for the gradient and the hessian f = x T Ax + b T x x f = f x = (A + AT )x + b 2 f xx T = A + AT Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 0

11 2.4 Derivatives of Traces 2 DERIVATIVES 2.4 Derivatives of Traces 2.4. First Order Second Order X Tr(X) = I X Tr(XB) = BT (6) X Tr(BXC) = BT C T X Tr(BXT C) = CB X Tr(XT C) = C X Tr(BXT ) = B See 7. X Tr(X2 ) = 2X X Tr(X2 B) = (XB + BX) T X Tr(XT BX) = BX + B T X X Tr(XBXT ) = XB T + XB X Tr(XT X) = 2X X Tr(BXXT ) = (B + B T )X X Tr(BT X T CXB) = C T XBB T + CXBB T X Tr X T BXC = BXC + B T XC T X Tr(AXBXT C) = A T C T XB T + CAXB X Tr (AXb + c)(axb + c) T = 2A T (AXb + c)b T Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page

12 2.5 Derivatives of Structured Matrices 2 DERIVATIVES Higher Order X Tr(Xk ) = k(x k ) T k X Tr(AXk ) = (X r AX k r ) T r=0 X Tr B T X T CXX T CXB = CXX T CXBB T + C T XBB T X T C T X Other + CXBB T X T CX + C T XX T C T XBB T X Tr(AX B) = (X BAX ) T = X T A T B T X T Assume B and C to be symmetric, then X Tr (X T CX) A = (CX(X T CX) )(A + A T )(X T CX) X Tr (X T CX) (X T BX) = 2CX(X T CX) X T BX(X T CX) +2BX(X T CX) See Derivatives of Structured Matrices Assume that the matrix A has some structure, i.e. is symmetric, toeplitz, etc. In that case the derivatives of the previous section does not apply in general. In stead, consider the following general rule for differentiating a scalar function f(a) df = T f A kl f A = Tr da ij A kl A ij A A ij kl 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 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. Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 2

13 2.5 Derivatives of Structured Matrices 2 DERIVATIVES 2.5. Symmetric If A is symmetric, then S ij = J ij + J ji J ij J ij and therefore That is, e.g., (5, 6): T df f f f da = + diag A A A Tr(AX) X det(x) X ln det(x) X = A + A T (A I), see (20) (7) = 2X (X I) (8) = 2X (X I) (9) Diagonal If X is diagonal, then (0): Tr(AX) X = A I (20) Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 3

14 3 INVERSES 3 Inverses 3. Exact Relations 3.. The Woodbury identity (A + CBC T ) = A A C(B + C T A C) C T A If P, R are positive definite, then (see 7) (P + B T R B) B T R = PB T (BPB T + R) 3..2 The Kailath Variant (A + BC) = A A B(I + CA B) CA See 4 page The Searle Set of Identities The following set of identities, can be found in 3, page 5, (I + A ) = A(A + I) (A + BB T ) B = A B(I + B T A B) (A + B ) = A(A + B) B = B(A + B) A A A(A + B) A = B B(A + B) B A + B = A (A + B)B (I + AB) = I A(I + BA) B (I + AB) A = A(I + AB) 3.2 Implication on Inverses See 3. (A + B) = A + B AB A = BA B 3.2. A PosDef identity Assume P, R to be positive definite and invertible, then (P + B T R B) B T R = PB T (BPB T + R) See?. Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 4

15 3.3 Approximations 3 INVERSES 3.3 Approximations If σ 2 is small then (I + A) = I A + A 2 A A A(I + A) A = I A 3.4 Generalized Inverse 3.4. Definition if A large and symmetric (Q + σ 2 M) = Q σ 2 Q MQ A generalized inverse matrix of the matrix A is any matrix A such that The matrix A is not unique. 3.5 Pseudo Inverse 3.5. Definition AA A = A The pseudo inverse (or Moore-Penrose 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 Properties Assume A + to be the pseudo-inverse of A, then (See 3) (A + ) + = A (A T ) + = (A + ) T (ca) + = (/c)a + (A T A) + = A + (A T ) + (AA T ) + = (A T ) + A + Assume A to have full rank, then (AA + )(AA + ) = AA + (A + A)(A + A) = A + A Tr(AA + ) = rank(aa + ) (See 4) Tr(A + A) = rank(a + A) (See 4) Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 5

16 3.5 Pseudo Inverse 3 INVERSES Construction Assume that A has full rank, then A n n Square rank(a) = n A + = A A n m Broad rank(a) = n A + = A T (AA T ) A n m Tall rank(a) = m A + = (A T A) A T 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 + = VD + U T A different way is this: There does always exists 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 ) (C T C) C T Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 6

17 4 COMPLEX MATRICES 4 Complex Matrices 4. Complex Derivatives In order to differentiate an expression f(z) with respect to a complex z, the Cauchy-Riemann 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 + i I(f(z)) Rz + I(f(z)) Iz (2) (22) f(z) Iz = if(z) Rz. (23) A complex function that satisfies the Cauchy-Riemann 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 Cauchy-Riemann equations. In order to avoid this problem, a more generalized definition of complex derivative is used (2, 6): Generalized Complex Derivative: df(z) dz = 2 Conjugate Complex Derivative ( f(z) ) Rz if(z) Iz df(z) dz = ( f(z) ) 2 Rz + if(z) Iz (24) (25) The Generalized Complex Derivative equals the normal derivative, when f is an analytic function. For a non-analytic 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 when deriving a complex gradient. Notice: df(z) dz f(z) Rz + if(z) Iz (26) Complex Gradient Vector: If f is a real function of a complex vector z, then the complex gradient vector is given by (9, p. 798) f(z) = 2 df(z) dz (27) = f(z) Rz + if(z) Iz Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 7

18 4. 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 (2) f(z) = 2 df(z) dz (28) = f(z) RZ + if(z) IZ These expressions can be used for gradient descent algorithms. 4.. The Chain Rule for complex numbers The chain rule is a little more complicated when the function of a complex u = f(x) is non-analytic. For a non-analytic function, the following chain rule can be applied (?) g(u) x = g u u x + g u u x = g u ( g ) u u x + u x (29) Notice, if the function is analytic, the second term reduces to zero, and the function is reduced to the normal well-known 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..2 Complex Derivatives of Traces Tr(( g(u) U ) T U ) X. (30) 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 (3) = I (32) Since the two results have the same sign, the conjugate complex derivative (25) should be used. Tr(X) RX = Tr(XT ) RX i Tr(X) IX = ) itr(xt IX = I (33) = I (34) Here, the two results have different signs, the generalized complex derivative (24) should be used. Hereby, it can be seen that (??) holds even if X is a Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 8

19 4. Complex Derivatives 4 COMPLEX MATRICES complex number. Tr(AX H ) RX i Tr(AXH ) IX Tr(AX ) RX i Tr(AX ) IX = A (35) = A (36) = A T (37) = A T (38) Tr(XX H ) RX i Tr(XXH ) IX = Tr(XH X) RX = i Tr(XH X) IX By inserting (39) and (40) in (24) and (25), it can be seen that = 2RX (39) = i2ix (40) Tr(XX H ) = X X (4) Tr(XX H ) X = X (42) Since the function Tr(XX H ) is a real function of the complex matrix X, the complex gradient matrix (28) is given by Tr(XX H ) = 2 Tr(XXH ) X = 2X (43) 4..3 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 (9) and (5), det(x H AX) can be calculated as (see Sec. A.0.2 for details) det(x H AX) X and the complex conjugate derivative yields = ( det(x H AX) i det(xh AX) ) 2 RX IX = det(x H AX) ( (X H AX) X H A ) T (44) det(x H AX) X = ( det(x H AX) + i det(xh AX) ) 2 RX IX = det(x H AX)AX(X H AX) (45) Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 9

20 5 DECOMPOSITIONS 5 Decompositions 5. Eigenvalues and Eigenvectors 5.. Definition The eigenvectors v and eigenvalues λ are the ones satisfying Av i = λ i v i AV = VD, (D) ij = δ ij λ i where the columns of V are the vectors v i 5..2 General Properties 5..3 Symmetric eig(ab) = eig(ba) A is n m At most min(n, m) distinct λ i rank(a) = r At most r non-zero λ i Assume A is symmetric, then VV T = I (i.e. V is orthogonal) λ i R (i.e. λ i is real) Tr(A p ) = i λp i eig(i + ca) = + cλ i eig(a ) = λ i 5.2 Singular Value Decomposition Any n m matrix A can be written as where A = UDV T U = eigenvectors of AA T n n D = diag(eig(aa T )) n m V = eigenvectors of A T A m m 5.2. Symmetric Square decomposed into squares Assume A to be n n and symmetric. Then A = V D V T where D is diagonal with the eigenvalues of A and V is orthogonal and the eigenvectors of A. Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 20

21 5.3 Triangular Decomposition 5 DECOMPOSITIONS Square decomposed into squares Assume A to be n n. Then A = V D U T where D is diagonal with the square root of the eigenvalues of AA T, V is the eigenvectors of AA T and U T is the eigenvectors of A T A Square decomposed into rectangular Assume V D U T = 0 then we can expand the SVD of A into D 0 U T A = V V 0 D where the SVD of A is A = VDU T Rectangular decomposition I U T Assume A is n m A = V D U T where D is diagonal with the square root of the eigenvalues of AA T, V is the eigenvectors of AA T and U T is the eigenvectors of A T A Rectangular decomposition II Assume A is n m A = V D U T Rectangular decomposition III Assume A is n m A = V D U T where D is diagonal with the square root of the eigenvalues of AA T, V is the eigenvectors of AA T and U T is the eigenvectors of A T A. 5.3 Triangular Decomposition 5.3. Cholesky-decomposition Assume A is positive definite, then A = B T B where B is a unique upper triangular matrix. Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 2

22 6 GENERAL STATISTICS AND PROBABILITY 6 General Statistics and Probability 6. Moments of any distribution 6.. Mean and covariance of linear forms Assume X and x to be a matrix and a vector of random variables. Then EAXB + C = AEXB + C See 4. VarAx = AVarxA T CovAx, By = ACovx, yb T 6..2 Mean and Variance of Square Forms Assume A is symmetric, c = Ex and Σ = Varx. Assume also that all coordinates x i are independent, have the same central moments µ, µ 2, µ 3, µ 4 and denote a = diag(a). Then See 4 Ex T Ax = Tr(AΣ) + c T Ac Varx T Ax = 2µ 2 2Tr(A 2 ) + 4µ 2 c T A 2 c + 4µ 3 c T Aa + (µ 4 3µ 2 2)a T a 6.2 Expectations Assume x to be a stochastic vector with mean m, covariance M and central moments v r = E(x m) r Linear Forms Quadratic Forms EAx + b = Am + b EAx = Am Ex + b = m + b E(Ax + a)(bx + b) T = AMB T + (Am + a)(bm + b) T Exx T = M + mm T Exa T x = (M + mm T )a Ex T ax T = a T (M + mm T ) E(Ax)(Ax) T = A(M + mm T )A T E(x + a)(x + a) T = M + (m + a)(m + a) T Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 22

23 6.2 Expectations 6 GENERAL STATISTICS AND PROBABILITY E(Ax + a) T (Bx + b) = Tr(AMB T ) + (Am + a) T (Bm + b) Ex T x = Tr(M) + m T m Ex T Ax = Tr(AM) + m T Am E(Ax) T (Ax) = Tr(AMA T ) + (Am) T (Am) E(x + a) T (x + a) = Tr(M) + (m + a) T (m + a) See Cubic Forms Assume x to be independent, then E(Ax + a)(bx + b) T (Cx + c) = Adiag(B T C)v 3 +Tr(BMC T )(Am + a) +AMC T (Bm + b) +(AMB T + (Am + a)(bm + b) T )(Cm + c) Exx T x = v 3 + 2Mm + (Tr(M) + m T m)m E(Ax + a)(ax + a) T (Ax + a) = Adiag(A T A)v 3 +2AMA T + (Ax + a)(ax + a) T (Am + a) +Tr(AMA T )(Am + a) E(Ax + a)b T (Cx + c)(dx + d) T = (Ax + a)b T (CMD T + (Cm + c)(dm + d) T ) +(AMC T + (Am + a)(cm + c) T )b(dm + d) T +b T (Cm + c)(amd T (Am + a)(dm + d) T ) See 7. Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 23

24 7 GAUSSIANS 7 Gaussians 7. One Dimensional 7.. Density and Normalization The density is Normalization integrals p(s) = ( ) exp (s µ)2 2πσ 2 2σ 2 e (s µ) 2 2σ 2 ds = 2πσ 2 π b e (ax2 +bx+c) 2 dx = a exp 4ac 4a e c 2x 2 +c x+c 0 π c 2 dx = exp 4c 2 c 0 c 2 4c Completing the Squares c 2 x 2 + c x + c 0 = a(x b) 2 + w or a = c 2 b = c w = 2 c 2 4 c 2 c 2 + c 0 c 2 x 2 + c x + c 0 = 2σ 2 (x µ)2 + d µ = c 2c 2 σ 2 = 2c 2 d = c 0 c2 4c Moments If the density is expressed by p(x) = exp (s µ)2 2πσ 2 2σ 2 or p(x) = C exp(c 2 x 2 + c x) then the first few basic moments are x = µ = c 2c 2 x 2 = σ 2 + µ 2 = 2c 2 + x 3 = 3σ 2 µ + µ 3 = x 4 = µ 4 + 6µ 2 σ 2 + 3σ 4 = and the central moments are c (2c 2 ) 2 ( ) 2 c 2c 2 3 c2 ( ) 4 ( c 2c c 2c 2 ) 2 ( ) ( 2c 2 2c ) 2 2c 2 Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 24

25 7.2 Basics 7 GAUSSIANS (x µ) = 0 = 0 (x µ) 2 = σ 2 = 2c 2 (x µ) 3 = 0 = 0 (x µ) 4 = 3σ 4 = 3 2 2c 2 A kind of pseudo-moments (un-normalized integrals) can easily be derived as π c exp(c 2 x 2 + c x)x n dx = Z x n 2 = exp x n c 2 4c 2 From the un-centralized moments one can derive other entities like 7.2 Basics x 2 x 2 = σ 2 = 2c 2 x 3 x 2 x = 2σ 2 µ = x 4 x 2 2 = 2σ 4 + 4µ 2 σ 2 = Density and normalization 2c (2c 2) 2 (2c 2) 2 4 c2 The density of x N (m, Σ) is p(x) = exp det(2πσ) 2 (x m)t Σ (x m) 2c 2 Note that if x is d-dimensional, then det(2πσ) = (2π) d det(σ). Integration and normalization exp 2 (x m)t Σ (x m) dx = det(2πσ) exp 2 xt Ax + b T x dx = det(2πa ) exp exp 2 Tr(ST AS) + Tr(B T S) ds = det(2πa ) exp The derivatives of the density are p(x) x = p(x)σ (x m) 2 p ( xx T = p(x) Σ (x m)(x m) T Σ Σ ) Linear combination Assume x N (m x, Σ x ) and y N (m y, Σ y ) then 2 bt A b 2 Tr(BT A B) Ax + By + c N (Am x + Bm y + c, AΣ x A T + BΣ y B T ) Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 25

26 7.2 Basics 7 GAUSSIANS Rearranging Means N Ax m, Σ = det(2π(at Σ A) ) det(2πσ) N x A m, (A T Σ A) Rearranging into squared form If A is symmetric, then 2 xt Ax + b T x = 2 (x A b) T A(x A b) + 2 bt A b 2 Tr(XT AX)+Tr(B T X) = 2 Tr(X A B) T A(X A B)+ 2 Tr(BT A B) Sum of two squared forms In vector formulation (assuming Σ, Σ 2 are symmetric) 2 (x m ) T Σ (x m ) 2 (x m 2) T Σ 2 (x m 2) = 2 (x m c) T Σ c (x m c ) + C Σ c = Σ + Σ 2 m c = (Σ + Σ 2 ) (Σ m + Σ 2 m 2) C = 2 (mt Σ + m T 2 Σ 2 )(Σ + Σ ( ) m T Σ 2 m + m T 2 Σ 2 m 2 2 ) (Σ In a trace formulation (assuming Σ, Σ 2 are symmetric) 2 Tr((X M ) T Σ (X M )) 2 Tr((X M 2) T Σ 2 (X M 2)) = 2 Tr(X M c) T Σ c (X M c ) + C m + Σ 2 m 2) Σ c = Σ + Σ 2 M c = (Σ + Σ 2 ) (Σ M + Σ 2 M 2) C = 2 Tr (Σ M + Σ 2 M 2) T (Σ + Σ 2 ) (Σ M + Σ 2 M 2) 2 Tr(MT Σ M + M T 2 Σ 2 M 2) Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 26

27 7.3 Moments 7 GAUSSIANS Product of gaussian densities Let N x (m, Σ) denote a density of x, then N x (m, Σ ) N x (m 2, Σ 2 ) = c c N x (m c, Σ c ) c c = N m (m 2, (Σ + Σ 2 )) = det(2π(σ + Σ 2 )) exp 2 (m m 2 ) T (Σ + Σ 2 ) (m m 2 ) m c = (Σ + Σ 2 ) (Σ m + Σ 2 m 2) Σ c = (Σ + Σ 2 ) but note that the product is not normalized as a density of x. 7.3 Moments 7.3. Mean and covariance of linear forms First and second moments. Assume x N (m, Σ) E(x) = m Cov(x, x) = Var(x) = Σ = E(xx T ) E(x)E(x T ) = E(xx T ) mm T As for any other distribution is holds for gaussians that EAx = AEx VarAx = AVarxA T CovAx, By = ACovx, yb T Mean and variance of square forms Mean and variance of square forms: Assume x N (m, Σ) E(xx T ) = Σ + mm T Ex T Ax = Tr(AΣ) + m T Am Var(x T Ax) = 2σ 4 Tr(A 2 ) + 4σ 2 m T A 2 m E(x m ) T A(x m ) = (m m ) T A(m m ) + Tr(AΣ) Assume x N (0, σ 2 I) and A and B to be symmetric, then Cov(x T Ax, x T Bx) = 2σ 4 Tr(AB) Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 27

28 7.3 Moments 7 GAUSSIANS Cubic forms Exb T xx T = mb T (M + mm T ) + (M + mm T )bm T Mean of Quartic Forms +b T m(m mm T ) Exx T xx T = 2(Σ + mm T ) 2 + m T m(σ mm T ) +Tr(Σ)(Σ + mm T ) Exx T Axx T = (Σ + mm T )(A + A T )(Σ + mm T ) +m T Am(Σ mm T ) + TrAΣ(Σ + mm T ) Ex T xx T x = 2Tr(Σ 2 ) + 4m T Σm + (Tr(Σ) + m T m) 2 Ex T Axx T Bx = TrAΣ(B + B T )Σ + m T (A + A T )Σ(B + B T )m Ea T xb T xc T xd T x +(Tr(AΣ) + m T Am)(Tr(BΣ) + m T Bm) = (a T (Σ + mm T )b)(c T (Σ + mm T )d) +(a T (Σ + mm T )c)(b T (Σ + mm T )d) +(a T (Σ + mm T )d)(b T (Σ + mm T )c) 2a T mb T mc T md T m E(Ax + a)(bx + b) T (Cx + c)(dx + d) T = AΣB T + (Am + a)(bm + b) T CΣD T + (Cm + c)(dm + d) T +AΣC T + (Am + a)(cm + c) T BΣD T + (Bm + b)(dm + d) T +(Bm + b) T (Cm + c)aσd T (Am + a)(dm + d) T +Tr(BΣC T )AΣD T + (Am + a)(dm + d) T E(Ax + a) T (Bx + b)(cx + c) T (Dx + d) = TrAΣ(C T D + D T C)ΣB T See 7. +(Am + a) T B + (Bm + b) T AΣC T (Dm + d) + D T (Cm + c) +Tr(AΣB T ) + (Am + a) T (Bm + b)tr(cσd T ) + (Cm + c) T (Dm + d) Moments Cov(x) = k Ex = ρ k m k k ρ k ρ k (Σ k + m k m T k m k m T k ) k Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 28

29 7.4 Miscellaneous 7 GAUSSIANS 7.4 Miscellaneous 7.4. Whitening Assume x N (m, Σ) then z = Σ /2 (x m) N (0, I) Conversely having z N (0, I) one can generate data x N (m, Σ) by setting x = Σ /2 z + m N (m, Σ) Note that Σ /2 means the matrix which fulfils Σ /2 Σ /2 = Σ, and that it exists and is unique since Σ is positive definite The Chi-Square connection Assume x N (m, Σ) and x to be n dimensional, then Entropy z = (x m) T Σ (x m) χ 2 n Entropy of a D-dimensional gaussian H(x) = N (m, Σ) ln N (m, Σ)dx = ln det(2πσ) D One Dimensional Mixture of Gaussians 7.5. Density and Normalization p(s) = K k ρ k exp 2πσ 2 k 2 (s µ k ) 2 σ 2 k Moments An useful fact of MoG, is that x n = k ρ k x n k where k denotes average with respect to the k.th component. We can calculate the first four moments from the densities p(x) = ρ k exp (x µ k ) 2 k 2πσ 2 k 2 σk 2 p(x) = ρ k C k exp c k2 x 2 + c k x k as Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 29

30 7.6 Mixture of Gaussians 7 GAUSSIANS x = k ρ kµ k = k ρ ck k x 2 = k ρ k(σk 2 + µ2 k ) = ( ) 2 k ρ k 2c k2 + ck 2c k2 x 3 = k ρ k(3σk 2µ k + µ 3 k ) = k ρ ck k (2c k2 ) 3 c2 2 k 2c k2 2c k2 x 4 = k ρ k(µ 4 k + 6µ2 k σ2 k + 3σ4 k ) = k ρ k ( 2c k2 ) 2 ( ck 2c k2 ) 2 6 c 2 k 2c k2 + 3 If all the gaussians are centered, i.e. µ k = 0 for all k, then x = 0 = 0 x 2 = k ρ kσk 2 = k ρ k x 3 = 0 = 0 x 4 = k ρ k3σk 4 = k ρ k3 2c k2 2 2c k2 From the un-centralized moments one can derive other entities like x 2 x 2 = k,k ρ kρ k µ 2 k + σk 2 µ kµ k x 3 x 2 x = k,k ρ kρ k 3σ 2 k µ k + µ 3 k (σ2 k + µ2 k )µ k x 4 x 2 2 = k,k ρ kρ k µ 4 k + 6µ 2 k σ2 k + 3σ4 k (σ2 k + µ2 k )(σ2 k + µ2 k ) 7.6 Mixture of Gaussians 7.6. Density The variable x is distributed as a mixture of gaussians if it has the density p(x) = K k= ρ k det(2πσk ) exp 2 (x m k) T Σ k (x m k) where ρ k sum to and the Σ k all are positive definite. Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 30

31 8 MISCELLANEOUS 8 Miscellaneous 8. Functions and Series 8.. Finite Series (X n I)(X I) = I + X + X X n 8..2 Taylor Expansion of Scalar Function Consider some scalar function f(x) which takes the vector x as an argument. This we can Taylor expand around x 0 f(x) = f(x 0 ) + g(x 0 ) T (x x 0 ) + 2 (x x 0) T H(x 0 )(x x 0 ) where g(x 0 ) = f(x) x x0 H(x 0 ) = 2 f(x) xx T x Taylor Expansion of Vector Functions 8..4 Matrix Functions by Infinite Series As for analytical functions in one dimension, one can define a matrix function for square matrices X by an infinite series f(x) = c n X n n=0 assuming the limit exists and is finite. If the coefficients c n fulfils n c nx n <, then one can prove that the above series exists and is finite, see. Thus for any analytical function f(x) there exists a corresponding matrix function f(x) constructed by the Taylor expansion. Using this one can prove the following results: ) A matrix A is a zero of its own characteristic polynomium : p(λ) = det(iλ A) = n c n λ n p(a) = 0 2) If A is square it holds that A = UBU f(a) = Uf(B)U 3) A useful fact when using power series is that A n 0 for n if A < Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 3

32 8.2 Indices, Entries and Vectors 8 MISCELLANEOUS 8..5 Exponential Matrix Function In analogy to the ordinary scalar exponential function, one can define exponential and logarithmic matrix functions: e A = e A = e ta = ln(i + A) = n=0 n=0 n=0 n! An = I + A + 2 A n! ( )n A n = I A + 2 A2... n! (ta)n = I + ta + 2 t2 A ( ) n A n = A n 2 A2 + 3 A3... n= Some of the properties of the exponential function are e A e B = e A+B if AB = BA (e A ) = e A d dt eta = Ae ta, 8..6 Trigonometric Functions sin(a) = cos(a) = n=0 ( ) n A 2n+ (2n + )! t R = A 3! A3 + 5! A5... ( ) n A 2n = I (2n)! 2! A2 + 4! A4... n=0 8.2 Indices, Entries and Vectors Let e i denote the column vector which is on entry i and zero elsewhere, i.e. (e i ) j = δ ij, and let J ij denote the matrix which is on entry (i, j) and zero elsewhere Rows and Columns i.th row of A = e T i A j.th column of A = Ae j Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 32

33 8.2 Indices, Entries and Vectors 8 MISCELLANEOUS Permutations Let P be some permutation matrix, e.g. 0 0 P = = e 2 e e 3 = e T 2 e T e T 3 then AP = Ae 2 Ae Ae 2 PA = e T 2 A e T A e T 3 A That is, the first is a matrix which has columns of A but in permuted sequence and the second is a matrix which has the rows of A but in the permuted sequence Swap and Zeros Assume A to be n m and J ij to be m p AJ ij = A i... 0 i.e. an n p matrix of zeros with the i.th column of A in the placed of the j.th column. Assume A to be n m and J ij to be p n 0 0 J ij A =. A j.. 0 i.e. an p m matrix of zeros with the j.th row of A in the placed of the i.th row Rewriting product of elements A ki B jl = (Ae i e T j B) kl = (AJ ij B) kl A ik B lj = (A T e i e T j B T ) kl = (A T J ij B T ) kl A ik B jl = (A T e i e T j B) kl = (A T J ij B) kl A ki B lj = (Ae i e T j B T ) kl = (AJ ij B T ) kl Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 33

34 8.2 Indices, Entries and Vectors 8 MISCELLANEOUS Properties of the Singleentry Matrix If i = j If i j J ij J ij = J ij (J ij ) T (J ij ) T = J ij J ij (J ij ) T = J ij (J ij ) T J ij = J ij J ij J ij = 0 (J ij ) T (J ij ) T = 0 J ij (J ij ) T = J ii (J ij ) T J ij = J jj The Singleentry Matrix in Scalar Expressions Assume A is n m and J is m n, then Tr(AJ ij ) = Tr(J ij A) = (A T ) ij Assume A is n n, J is n m and B is m n, then Tr(AJ ij B) = (A T B T ) ij Tr(AJ ji B) = (BA) ij Tr(AJ ij J ij B) = diag(a T B T ) ij Assume A is n n, J ij is n m B is m n, then Structure Matrices The structure matrix is defined by If A has no special structure then x T AJ ij Bx = (A T xx T B T ) ij x T AJ ij J ij Bx = diag(a T xx T B T ) ij A A ij = S ij S ij = J ij If A is symmetric then S ij = J ij + J ji J ij J ij Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 34

35 8.3 Solutions to Systems of Equations 8 MISCELLANEOUS 8.3 Solutions to Systems of Equations 8.3. Existence in Linear Systems Assume A is n m and consider the linear system Ax = b Construct the augmented matrix B = A b then Condition rank(a) = rank(b) = m rank(a) = rank(b) < m rank(a) < rank(b) Solution Unique solution x Many solutions x No solutions x Standard Square Assume A is square and invertible, then Degenerated Square Ax = b x = A b Over-determined Rectangular Assume A to be n m, n > m (tall) and rank(a) = m, then Ax = b x = (A T A) A T b = A + b that is if there exists a solution x at all! If there is no solution the following can be useful: Ax = b x min = A + b Now x min is the vector x which minimizes Ax b 2, i.e. the vector which is least wrong. The matrix A + is the pseudo-inverse of A. See Under-determined Rectangular Assume A is n m and n < m ( broad ). Ax = b x min = A T (AA T ) b The equation have many solutions x. But x min is the solution which minimizes Ax b 2 and also the solution with the smallest norm x 2. The same holds for a matrix version: Assume A is n m, X is m n and B is n n, then AX = B X min = A + B The equation have many solutions X. But X min is the solution which minimizes AX B 2 and also the solution with the smallest norm X 2. See 3. Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 35

36 8.4 Block matrices 8 MISCELLANEOUS Similar but different: Assume A is square n n and the matrices B 0, B are n N, where N > n, then if B 0 has maximal rank AB 0 = B A min = B B T 0 (B 0 B T 0 ) where A min denotes the matrix which is optimal in a least square sense. An interpretation is that A is the linear approximation which maps the columns vectors of B 0 into the columns vectors of B Linear form and zeros Ax = 0, x A = Square form and zeros If A is symmetric, then x T Ax = 0, x A = Block matrices Let A ij denote the ij.th block of A Multiplication Assuming the dimensions of the blocks matches we have A A 2 B B 2 A B = + A 2 B 2 A B 2 + A 2 B 22 A 2 A 22 B 2 B 22 A 2 B + A 22 B 2 A 2 B 2 + A 22 B The Determinant The determinant can be expressed as by the use of C = A A 2 A 22 A 2 C 2 = A 22 A 2 A A 2 as ( A A det 2 A 2 A 22 ) = det(a 22 ) det(c ) = det(a ) det(c 2 ) The Inverse The inverse can be expressed as by the use of C = A A 2 A 22 A 2 C 2 = A 22 A 2 A A 2 Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 36

37 8.5 Matrix Norms 8 MISCELLANEOUS as A A 2 = = C A A 2C 2 C 2 A 2A C 2 A 2 A 22 A + A A 2C 2 A 2A C A 2A Block diagonal A 22 A 2C A 22 + A 22 A 2C A 2A 22 For block diagonal matrices we have A 0 0 A 22 = ( A 0 det 0 A Matrix Norms 8.5. Definitions A matrix norm is a mapping which fulfils Examples (A ) 0 0 (A 22 ) ) = det(a ) det(a 22 ) A 0 A = 0 A = 0 ca = c A, c R A + B A + B A = max A ij j i A 2 = max eig(a T A) A p = ( max Ax p) /p x p= A = max A ij i j A F = A ij 2 (Frobenius) ij A max = max A ij ij A KF = sing(a) (Ky Fan) where sing(a) is the vector of singular values of the matrix A. Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 37

38 8.6 Positive Definite and Semi-definite Matrices 8 MISCELLANEOUS Inequalities E. H. Rasmussen has in yet unpublished material derived and collected the following inequalities. They are collected in a table as below, assuming A is an m n, and d = min{m, n} A max A A A 2 A F A KF A max A m m m m m A n n n n n A 2 mn n m A F mn n m d A KF mnd nd md d d which are to be read as, e.g. A 2 m A 8.6 Positive Definite and Semi-definite Matrices 8.6. Definitions A matrix A is positive definite if and only if x T Ax > 0, x A matrix A is positive semi-definite if and only if x T Ax 0, x Note that if A is positive definite, then A is also positive semi-definite Eigenvalues The following holds with respect to the eigenvalues: Trace A pos. def. eig(a) > 0 A pos. semi-def. eig(a) 0 The following holds with respect to the trace: Inverse A pos. def. Tr(A) > 0 A pos. semi-def. Tr(A) 0 If A is positive definite, then A is invertible and A is also positive definite. Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 38

39 8.7 Integral Involving Dirac Delta Functions 8 MISCELLANEOUS Diagonal If A is positive definite, then A ii > 0, i Decomposition I The matrix A is positive semi-definite of rank r there exists a matrix B of rank r such that A = BB T The matrix A is positive definite there exists an invertible matrix B such that A = BB T Decomposition II Assume A is an n n positive semi-definite, then there exists an n r matrix B of rank r such that B T AB = I Equation with zeros Assume A is positive semi-definite, then X T AX = 0 AX = Rank of product Assume A is positive definite, then rank(bab T ) = rank(b) Positive definite property If A is n n positive definite and B is r n of rank r, then BAB T is positive definite Outer Product If X is n r of rank r, then XX T is positive definite Small pertubations If A is positive definite and B is symmetric, then A tb is positive definite for sufficiently small t. 8.7 Integral Involving Dirac Delta Functions Assuming A to be square, then p(s)δ(x As)ds = det(a) p(a x) Assuming A to be underdetermined, i.e. tall, then { p(s)δ(x As)ds = det(a T A) p(a+ x) if x = AA + x 0 elsewhere } Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 39

40 8.8 Miscellaneous 8 MISCELLANEOUS See Miscellaneous For any A it holds that rank(a) = rank(a T ) = rank(aa T ) = rank(a T A) Assume A is positive definite. Then rank(b T AB) = rank(b) A is positive definite B invertible, such that A = BB T Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 40

41 A PROOFS AND DETAILS A Proofs and Details A.0. Proof of Equation 4 Essentially we need to calculate (X n ) kl X ij = X ij u,...,u n X k,u X u,u 2...X un,l = δ k,i δ u,jx u,u 2...X un,l.. +X k,u δ u,iδ u2,j...x un,l = = +X k,u X u,u 2...δ un,iδ l,j n (X r ) ki (X n r ) jl r=0 n (X r J ij X n r ) kl r=0 Using the properties of the single entry matrix found in Sec , the result follows easily. A.0.2 Details on Eq. 47 det(x H AX) = det(x H AX)Tr(X H AX) (X H AX) = det(x H AX)Tr(X H AX) ((X H )AX + X H (AX)) = det(x H AX) ( Tr(X H AX) (X H )AX + Tr(X H AX) X H (AX) ) = det(x H AX) ( TrAX(X H AX) (X H ) + Tr(X H AX) X H A(X) ) First, the derivative is found with respect to the real part of X det(x H AX) RX ( TrAX(X = det(x H H AX) (X H ) AX) + Tr(XH AX) X H A(X) ) RX RX = det(x H AX) ( AX(X H AX) + ((X H AX) X H A) T ) Through the calculations, (6) and (35) were used. In addition, by use of (36), the derivative is found with respect to the imaginary part of X i det(xh AX) IX ( TrAX(X = i det(x H H AX) (X H ) AX) + Tr(XH AX) X H A(X) ) IX IX = det(x H AX) ( AX(X H AX) ((X H AX) X H A) T ) Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 4

42 A PROOFS AND DETAILS Hence, derivative yields det(x H AX) X = ( det(x H AX) i det(xh AX) ) 2 RX IX = det(x H AX) ( (X H AX) X H A ) T and the complex conjugate derivative yields det(x H AX) X = ( det(x H AX) + i det(xh AX) ) 2 RX IX = det(x H AX)AX(X H AX) Notice, for real X, A, the sum of (44) and (45) is reduced to (3). Similar calculations yield det(xax H ) X = ( det(xax H ) i det(xaxh ) ) 2 RX IX = det(xax H ) ( AX H (XAX H ) ) T (46) and det(xax H ) X = ( det(xax H ) + i det(xaxh ) ) 2 RX IX = det(xax H )(XAX H ) XA (47) Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 42

43 REFERENCES REFERENCES References Karl Gustav Andersson and Lars-Christer Boiers. Ordinaera differentialekvationer. Studenterlitteratur, Jörn Anemüller, Terrence J. Sejnowski, and Scott Makeig. Complex independent component analysis of frequency-domain electroencephalographic data. Neural Networks, 6(9):3 323, November S. Barnet. Matrices. Methods and Applications. Oxford Applied Mathematics and Computin Science Series. Clarendon Press, Christoffer Bishop. Neural Networks for Pattern Recognition. Oxford University Press, Robert J. Boik. Lecture notes: Statistics 550. Online, April Notes. 6 D. H. Brandwood. A complex gradient operator and its application in adaptive array theory. IEE Proceedings, 30(): 6, February 983. PTS. F and H. 7 M. Brookes. Matrix reference manual, Website May 20, Mads Dyrholm. Some matrix results, Website August 23, Simon Haykin. Adaptive Filter Theory. Prentice Hall, Upper Saddle River, NJ, 4th edition, Thomas P. Minka. Old and new matrix algebra useful for statistics, December Notes. L. Parra and C. Spence. Convolutive blind separation of non-stationary sources. In IEEE Transactions Speech and Audio Processing, pages , May Laurent Schwartz. Cours d Analyse, volume II. Hermann, Paris, 967. As referenced in 9. 3 Shayle R. Searle. Matrix Algebra Useful for Statistics. John Wiley and Sons, G. Seber and A. Lee. Linear Regression Analysis. John Wiley and Sons, S. M. Selby. Standard Mathematical Tables. CRC Press, Inna Stainvas. Matrix algebra in differential calculus. Neural Computing Research Group, Information Engeneering, Aston University, UK, August Notes. 7 Max Welling. The kalman filter. Lecture Note. Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 43