{"product_id":"matrix-differential-calculus-with-applications-in-statistics-and-econometrics-isbn-9781119541202","title":"Matrix Differential Calculus with Applications in Statistics and Econometrics","description":"\u003cp\u003e\u003cb\u003eA brand new, fully updated edition of a popular classic on matrix differential calculus with applications in statistics and econometrics\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eThis exhaustive, self-contained book on matrix theory and matrix differential calculus provides a treatment of matrix calculus based on differentials and shows how easy it is to use this theory once you have mastered the technique. Jan Magnus, who, along with the late Heinz Neudecker, pioneered the theory, develops it further in this new edition and provides many examples along the way to support it.\u003c\/p\u003e \u003cp\u003eMatrix calculus has become an essential tool for quantitative methods in a large number of applications, ranging from social and behavioral sciences to econometrics. It is still relevant and used today in a wide range of subjects such as the biosciences and psychology. \u003ci\u003eMatrix Differential Calculus with Applications in Statistics and Econometrics, Third Edition \u003c\/i\u003econtains all of the essentials of multivariable calculus with an emphasis on the use of differentials. It starts by presenting a concise, yet thorough overview of matrix algebra, then goes on to develop the theory of differentials. The rest of the text combines the theory and application of matrix differential calculus, providing the practitioner and researcher with both a quick review and a detailed reference.\u003c\/p\u003e \u003cul\u003e \u003cli\u003eFulfills the need for an updated and unified treatment of matrix differential calculus\u003c\/li\u003e \u003cli\u003eContains many new examples and exercises based on questions asked of the author over the years\u003c\/li\u003e \u003cli\u003eCovers new developments in field and features new applications\u003c\/li\u003e \u003cli\u003eWritten by a leading expert and pioneer of the theory\u003c\/li\u003e \u003cli\u003ePart of the Wiley Series in Probability and Statistics \u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003e\u003ci\u003eMatrix Differential Calculus With Applications in Statistics and Econometrics Third Edition \u003c\/i\u003eis an ideal text for graduate students and academics studying the subject, as well as for postgraduates and specialists working in biosciences and psychology.\u003c\/p\u003e \u003cp\u003ePreface xiii\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart One — Matrices\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Basic properties of vectors and matrices 3\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introduction 3\u003c\/p\u003e \u003cp\u003e2 Sets 3\u003c\/p\u003e \u003cp\u003e3 Matrices: addition and multiplication 4\u003c\/p\u003e \u003cp\u003e4 The transpose of a matrix 6\u003c\/p\u003e \u003cp\u003e5 Square matrices 6\u003c\/p\u003e \u003cp\u003e6 Linear forms and quadratic forms 7\u003c\/p\u003e \u003cp\u003e7 The rank of a matrix 9\u003c\/p\u003e \u003cp\u003e8 The inverse 10\u003c\/p\u003e \u003cp\u003e9 The determinant 10\u003c\/p\u003e \u003cp\u003e10 The trace 11\u003c\/p\u003e \u003cp\u003e11 Partitioned matrices 12\u003c\/p\u003e \u003cp\u003e12 Complex matrices 14\u003c\/p\u003e \u003cp\u003e13 Eigenvalues and eigenvectors 14\u003c\/p\u003e \u003cp\u003e14 Schur’s decomposition theorem 17\u003c\/p\u003e \u003cp\u003e15 The Jordan decomposition 18\u003c\/p\u003e \u003cp\u003e16 The singular-value decomposition 20\u003c\/p\u003e \u003cp\u003e17 Further results concerning eigenvalues 20\u003c\/p\u003e \u003cp\u003e18 Positive (semi)definite matrices 23\u003c\/p\u003e \u003cp\u003e19 Three further results for positive definite matrices 25\u003c\/p\u003e \u003cp\u003e20 A useful result 26\u003c\/p\u003e \u003cp\u003e21 Symmetric matrix functions 27\u003c\/p\u003e \u003cp\u003eMiscellaneous exercises\u003ci\u003e \u003c\/i\u003e28\u003c\/p\u003e \u003cp\u003eBibliographical notes\u003ci\u003e \u003c\/i\u003e30\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Kronecker products, vec operator, and Moore-Penrose inverse 31\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introduction 31\u003c\/p\u003e \u003cp\u003e2 The Kronecker product 31\u003c\/p\u003e \u003cp\u003e3 Eigenvalues of a Kronecker product 33\u003c\/p\u003e \u003cp\u003e4 The vec operator 34\u003c\/p\u003e \u003cp\u003e5 The Moore-Penrose (MP) inverse 36\u003c\/p\u003e \u003cp\u003e6 Existence and uniqueness of the MP inverse 37\u003c\/p\u003e \u003cp\u003e7 Some properties of the MP inverse 38\u003c\/p\u003e \u003cp\u003e8 Further properties 39\u003c\/p\u003e \u003cp\u003e9 The solution of linear equation systems 41\u003c\/p\u003e \u003cp\u003eMiscellaneous exercises\u003ci\u003e \u003c\/i\u003e43\u003c\/p\u003e \u003cp\u003eBibliographical notes\u003ci\u003e \u003c\/i\u003e45\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Miscellaneous matrix results 47\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introduction 47\u003c\/p\u003e \u003cp\u003e2 The adjoint matrix 47\u003c\/p\u003e \u003cp\u003e3 Proof of Theorem 3.1 49\u003c\/p\u003e \u003cp\u003e4 Bordered determinants 51\u003c\/p\u003e \u003cp\u003e5 The matrix equation\u003ci\u003e AX\u003c\/i\u003e = 0 51\u003c\/p\u003e \u003cp\u003e6 The Hadamard product 52\u003c\/p\u003e \u003cp\u003e7 The commutation matrix\u003ci\u003e K\u003csub\u003emn\u003c\/sub\u003e \u003c\/i\u003e54\u003c\/p\u003e \u003cp\u003e8 The duplication matrix\u003ci\u003e D\u003csub\u003en\u003c\/sub\u003e\u003c\/i\u003e 56\u003c\/p\u003e \u003cp\u003e9 Relationship between \u003ci\u003eD\u003csub\u003en+1 \u003c\/sub\u003e\u003c\/i\u003eand \u003ci\u003eD\u003csub\u003en\u003c\/sub\u003e\u003c\/i\u003e, I 58\u003c\/p\u003e \u003cp\u003e10 Relationship between \u003ci\u003eD\u003csub\u003en+1 \u003c\/sub\u003e\u003c\/i\u003eand \u003ci\u003eD\u003csub\u003en\u003c\/sub\u003e\u003c\/i\u003e, II 59\u003c\/p\u003e \u003cp\u003e11 Conditions for a quadratic form to be positive (negative) subject to linear constraints 60\u003c\/p\u003e \u003cp\u003e12 Necessary and sufficient conditions for\u003ci\u003e r(A : B) = r(A) + r(B) \u003c\/i\u003e63\u003c\/p\u003e \u003cp\u003e13 The bordered Gramian matrix 65\u003c\/p\u003e \u003cp\u003e14 The equations\u003ci\u003e X\u003csub\u003e1\u003c\/sub\u003eA + X\u003csub\u003e2\u003c\/sub\u003eB′ = G\u003csub\u003e1\u003c\/sub\u003e,X\u003csub\u003e1\u003c\/sub\u003eB = G\u003csub\u003e2\u003c\/sub\u003e \u003c\/i\u003e67\u003c\/p\u003e \u003cp\u003eMiscellaneous exercises\u003ci\u003e \u003c\/i\u003e69\u003c\/p\u003e \u003cp\u003eBibliographical notes\u003ci\u003e \u003c\/i\u003e70\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart Two — Differentials: the theory\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Mathematical preliminaries 73\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introduction 73\u003c\/p\u003e \u003cp\u003e2 Interior points and accumulation points 73\u003c\/p\u003e \u003cp\u003e3 Open and closed sets 75\u003c\/p\u003e \u003cp\u003e4 The Bolzano-Weierstrass theorem 77\u003c\/p\u003e \u003cp\u003e5 Functions 78\u003c\/p\u003e \u003cp\u003e6 The limit of a function 79\u003c\/p\u003e \u003cp\u003e7 Continuous functions and compactness 80\u003c\/p\u003e \u003cp\u003e8 Convex sets 81\u003c\/p\u003e \u003cp\u003e9 Convex and concave functions 83\u003c\/p\u003e \u003cp\u003eBibliographical notes\u003ci\u003e \u003c\/i\u003e86\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Differentials and differentiability 87\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introduction 87\u003c\/p\u003e \u003cp\u003e2 Continuity 88\u003c\/p\u003e \u003cp\u003e3 Differentiability and linear approximation 90\u003c\/p\u003e \u003cp\u003e4 The differential of a vector function 91\u003c\/p\u003e \u003cp\u003e5 Uniqueness of the differential 93\u003c\/p\u003e \u003cp\u003e6 Continuity of differentiable functions 94\u003c\/p\u003e \u003cp\u003e7 Partial derivatives 95\u003c\/p\u003e \u003cp\u003e8 The first identification theorem 96\u003c\/p\u003e \u003cp\u003e9 Existence of the differential, I 97\u003c\/p\u003e \u003cp\u003e10 Existence of the differential, II 99\u003c\/p\u003e \u003cp\u003e11 Continuous differentiability 100\u003c\/p\u003e \u003cp\u003e12 The chain rule 100\u003c\/p\u003e \u003cp\u003e13 Cauchy invariance 102\u003c\/p\u003e \u003cp\u003e14 The mean-value theorem for real-valued functions 103\u003c\/p\u003e \u003cp\u003e15 Differentiable matrix functions 104\u003c\/p\u003e \u003cp\u003e16 Some remarks on notation 106\u003c\/p\u003e \u003cp\u003e17 Complex differentiation 108\u003c\/p\u003e \u003cp\u003eMiscellaneous exercises\u003ci\u003e \u003c\/i\u003e110\u003c\/p\u003e \u003cp\u003eBibliographical notes\u003ci\u003e \u003c\/i\u003e110\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 The second differential 111\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introduction 111\u003c\/p\u003e \u003cp\u003e2 Second-order partial derivatives 111\u003c\/p\u003e \u003cp\u003e3 The Hessian matrix 112\u003c\/p\u003e \u003cp\u003e4 Twice differentiability and second-order approximation, I 113\u003c\/p\u003e \u003cp\u003e5 Definition of twice differentiability 114\u003c\/p\u003e \u003cp\u003e6 The second differential 115\u003c\/p\u003e \u003cp\u003e7 Symmetry of the Hessian matrix 117\u003c\/p\u003e \u003cp\u003e8 The second identification theorem 119\u003c\/p\u003e \u003cp\u003e9 Twice differentiability and second-order approximation, II 119\u003c\/p\u003e \u003cp\u003e10 Chain rule for Hessian matrices 121\u003c\/p\u003e \u003cp\u003e11 The analog for second differentials 123\u003c\/p\u003e \u003cp\u003e12 Taylor’s theorem for real-valued functions 124\u003c\/p\u003e \u003cp\u003e13 Higher-order differentials 125\u003c\/p\u003e \u003cp\u003e14 Real analytic functions 125\u003c\/p\u003e \u003cp\u003e15 Twice differentiable matrix functions 126\u003c\/p\u003e \u003cp\u003eBibliographical notes\u003ci\u003e \u003c\/i\u003e127\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Static optimization 129\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introduction 129\u003c\/p\u003e \u003cp\u003e2 Unconstrained optimization 130\u003c\/p\u003e \u003cp\u003e3 The existence of absolute extrema 131\u003c\/p\u003e \u003cp\u003e4 Necessary conditions for a local minimum 132\u003c\/p\u003e \u003cp\u003e5 Sufficient conditions for a local minimum: first-derivative test 134\u003c\/p\u003e \u003cp\u003e6 Sufficient conditions for a local minimum: second-derivative test 136\u003c\/p\u003e \u003cp\u003e7 Characterization of differentiable convex functions 138\u003c\/p\u003e \u003cp\u003e8 Characterization of twice differentiable convex functions 141\u003c\/p\u003e \u003cp\u003e9 Sufficient conditions for an absolute minimum 142\u003c\/p\u003e \u003cp\u003e10 Monotonic transformations 143\u003c\/p\u003e \u003cp\u003e11 Optimization subject to constraints 144\u003c\/p\u003e \u003cp\u003e12 Necessary conditions for a local minimum under constraints 145\u003c\/p\u003e \u003cp\u003e13 Sufficient conditions for a local minimum under constraints 149\u003c\/p\u003e \u003cp\u003e14 Sufficient conditions for an absolute minimum under constraints 154\u003c\/p\u003e \u003cp\u003e15 A note on constraints in matrix form 155\u003c\/p\u003e \u003cp\u003e16 Economic interpretation of Lagrange multipliers 155\u003c\/p\u003e \u003cp\u003eAppendix: the implicit function theorem 157\u003c\/p\u003e \u003cp\u003eBibliographical notes\u003ci\u003e \u003c\/i\u003e159\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart Three — Differentials: the practice\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Some important differentials 163\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introduction 163\u003c\/p\u003e \u003cp\u003e2 Fundamental rules of differential calculus 163\u003c\/p\u003e \u003cp\u003e3 The differential of a determinant 165\u003c\/p\u003e \u003cp\u003e4 The differential of an inverse 168\u003c\/p\u003e \u003cp\u003e5 Differential of the Moore-Penrose inverse 169\u003c\/p\u003e \u003cp\u003e6 The differential of the adjoint matrix 172\u003c\/p\u003e \u003cp\u003e7 On differentiating eigenvalues and eigenvectors 174\u003c\/p\u003e \u003cp\u003e8 The continuity of eigenprojections 176\u003c\/p\u003e \u003cp\u003e9 The differential of eigenvalues and eigenvectors: symmetric case 180\u003c\/p\u003e \u003cp\u003e10 Two alternative expressions for dλ 183\u003c\/p\u003e \u003cp\u003e11 Second differential of the eigenvalue function 185\u003c\/p\u003e \u003cp\u003eMiscellaneous exercises\u003ci\u003e \u003c\/i\u003e186\u003c\/p\u003e \u003cp\u003eBibliographical notes\u003ci\u003e \u003c\/i\u003e189\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 First-order differentials and Jacobian matrices 191\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introduction 191\u003c\/p\u003e \u003cp\u003e2 Classification 192\u003c\/p\u003e \u003cp\u003e3 Derisatives 192\u003c\/p\u003e \u003cp\u003e4 Derivatives 194\u003c\/p\u003e \u003cp\u003e5 Identification of Jacobian matrices 196\u003c\/p\u003e \u003cp\u003e6 The first identification table 197\u003c\/p\u003e \u003cp\u003e7 Partitioning of the derivative 197\u003c\/p\u003e \u003cp\u003e8 Scalar functions of a scalar 198\u003c\/p\u003e \u003cp\u003e9 Scalar functions of a vector 198\u003c\/p\u003e \u003cp\u003e10 Scalar functions of a matrix, I: trace 199\u003c\/p\u003e \u003cp\u003e11 Scalar functions of a matrix, II: determinant 201\u003c\/p\u003e \u003cp\u003e12 Scalar functions of a matrix, III: eigenvalue 202\u003c\/p\u003e \u003cp\u003e13 Two examples of vector functions 203\u003c\/p\u003e \u003cp\u003e14 Matrix functions 204\u003c\/p\u003e \u003cp\u003e15 Kronecker products 206\u003c\/p\u003e \u003cp\u003e16 Some other problems 208\u003c\/p\u003e \u003cp\u003e17 Jacobians of transformations 209\u003c\/p\u003e \u003cp\u003eBibliographical notes\u003ci\u003e \u003c\/i\u003e210\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Second-order differentials and Hessian matrices 211\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introduction 211\u003c\/p\u003e \u003cp\u003e2 The second identification table 211\u003c\/p\u003e \u003cp\u003e3 Linear and quadratic forms 212\u003c\/p\u003e \u003cp\u003e4 A useful theorem 213\u003c\/p\u003e \u003cp\u003e5 The determinant function 214\u003c\/p\u003e \u003cp\u003e6 The eigenvalue function 215\u003c\/p\u003e \u003cp\u003e7 Other examples 215\u003c\/p\u003e \u003cp\u003e8 Composite functions 217\u003c\/p\u003e \u003cp\u003e9 The eigenvector function 218\u003c\/p\u003e \u003cp\u003e10 Hessian of matrix functions, I 219\u003c\/p\u003e \u003cp\u003e11 Hessian of matrix functions, II 219\u003c\/p\u003e \u003cp\u003eMiscellaneous exercises\u003ci\u003e \u003c\/i\u003e220\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart Four — Inequalities\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Inequalities 225\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introduction 225\u003c\/p\u003e \u003cp\u003e2 The Cauchy-Schwarz inequality 226\u003c\/p\u003e \u003cp\u003e3 Matrix analogs of the Cauchy-Schwarz inequality 227\u003c\/p\u003e \u003cp\u003e4 The theorem of the arithmetic and geometric means 228\u003c\/p\u003e \u003cp\u003e5 The Rayleigh quotient 230\u003c\/p\u003e \u003cp\u003e6 Concavity of λ\u003ci\u003e\u003csub\u003e1 \u003c\/sub\u003e\u003c\/i\u003eand convexity of λ\u003ci\u003e\u003csub\u003en \u003c\/sub\u003e\u003c\/i\u003e232\u003c\/p\u003e \u003cp\u003e7 Variational description of eigenvalues 232\u003c\/p\u003e \u003cp\u003e8 Fischer’s min-max theorem 234\u003c\/p\u003e \u003cp\u003e9 Monotonicity of the eigenvalues 236\u003c\/p\u003e \u003cp\u003e10 The Poincar´e separation theorem 236\u003c\/p\u003e \u003cp\u003e11 Two corollaries of Poincar´e’s theorem 237\u003c\/p\u003e \u003cp\u003e12 Further consequences of the Poincar´e theorem 238\u003c\/p\u003e \u003cp\u003e13 Multiplicative version 239\u003c\/p\u003e \u003cp\u003e14 The maximum of a bilinear form 241\u003c\/p\u003e \u003cp\u003e15 Hadamard’s inequality 242\u003c\/p\u003e \u003cp\u003e16 An interlude: Karamata’s inequality 242\u003c\/p\u003e \u003cp\u003e17 Karamata’s inequality and eigenvalues 244\u003c\/p\u003e \u003cp\u003e18 An inequality concerning positive semidefinite matrices 245\u003c\/p\u003e \u003cp\u003e19 A representation theorem for ( ∑a\u003csup\u003ep\u003c\/sup\u003e\u003csub\u003ei\u003c\/sub\u003e )\u003csup\u003e1\/p\u003c\/sup\u003e 246\u003c\/p\u003e \u003cp\u003e20 A representation theorem for (tr\u003ci\u003eA\u003c\/i\u003e\u003csup\u003ep\u003c\/sup\u003e)\u003csup\u003e1\/p \u003c\/sup\u003e247\u003c\/p\u003e \u003cp\u003e21 Hölder’s inequality 248\u003c\/p\u003e \u003cp\u003e22 Concavity of log|A| 250\u003c\/p\u003e \u003cp\u003e23 Minkowski’s inequality 251\u003c\/p\u003e \u003cp\u003e24 Quasilinear representation of |A|\u003csup\u003e1\/n\u003c\/sup\u003e 253\u003c\/p\u003e \u003cp\u003e25 Minkowski’s determinant theorem 255\u003c\/p\u003e \u003cp\u003e26 Weighted means of order\u003ci\u003e p\u003c\/i\u003e 256\u003c\/p\u003e \u003cp\u003e27 Schlömilch’s inequality 258\u003c\/p\u003e \u003cp\u003e28 Curvature properties of\u003ci\u003e M\u003csub\u003ep\u003c\/sub\u003e\u003c\/i\u003e(\u003ci\u003ex\u003c\/i\u003e,\u003ci\u003e a\u003c\/i\u003e) 259\u003c\/p\u003e \u003cp\u003e29 Least squares 260\u003c\/p\u003e \u003cp\u003e30 Generalized least squares 261\u003c\/p\u003e \u003cp\u003e31 Restricted least squares 262\u003c\/p\u003e \u003cp\u003e32 Restricted least squares: matrix version 264\u003c\/p\u003e \u003cp\u003eMiscellaneous exercises\u003ci\u003e \u003c\/i\u003e265\u003c\/p\u003e \u003cp\u003eBibliographical notes\u003ci\u003e \u003c\/i\u003e269\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart Five — The linear model\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Statistical preliminaries 273\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introduction 273\u003c\/p\u003e \u003cp\u003e2 The cumulative distribution function 273\u003c\/p\u003e \u003cp\u003e3 The joint density function 274\u003c\/p\u003e \u003cp\u003e4 Expectations 274\u003c\/p\u003e \u003cp\u003e5 Variance and covariance 275\u003c\/p\u003e \u003cp\u003e6 Independence of two random variables 277\u003c\/p\u003e \u003cp\u003e7 Independence of n random variables 279\u003c\/p\u003e \u003cp\u003e8 Sampling 279\u003c\/p\u003e \u003cp\u003e9 The one-dimensional normal distribution 279\u003c\/p\u003e \u003cp\u003e10 The multivariate normal distribution 280\u003c\/p\u003e \u003cp\u003e11 Estimation 282\u003c\/p\u003e \u003cp\u003eMiscellaneous exercises\u003ci\u003e \u003c\/i\u003e282\u003c\/p\u003e \u003cp\u003eBibliographical notes\u003ci\u003e \u003c\/i\u003e283\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 The linear regression model 285\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introduction 285\u003c\/p\u003e \u003cp\u003e2 Affine minimum-trace unbiased estimation 286\u003c\/p\u003e \u003cp\u003e3 The Gauss-Markov theorem 287\u003c\/p\u003e \u003cp\u003e4 The method of least squares 290\u003c\/p\u003e \u003cp\u003e5 Aitken’s theorem 291\u003c\/p\u003e \u003cp\u003e6 Multicollinearity 293\u003c\/p\u003e \u003cp\u003e7 Estimable functions 295\u003c\/p\u003e \u003cp\u003e8 Linear constraints: the case\u003ci\u003e M(R′) \u003c\/i\u003e\u003ci\u003e⊂M(X\u003c\/i\u003e\u003ci\u003e′) \u003c\/i\u003e296\u003c\/p\u003e \u003cp\u003e9 Linear constraints: the general case 300\u003c\/p\u003e \u003cp\u003e10 Linear constraints: the case\u003ci\u003e M(R′) ∩M(X′) \u003c\/i\u003e= {0} 302\u003c\/p\u003e \u003cp\u003e11 A singular variance matrix: the case\u003ci\u003e M(X) \u003c\/i\u003e\u003ci\u003e⊂M(V ) \u003c\/i\u003e304\u003c\/p\u003e \u003cp\u003e12 A singular variance matrix: the case\u003ci\u003e r(X′V +X) = r(X) \u003c\/i\u003e305\u003c\/p\u003e \u003cp\u003e13 A singular variance matrix: the general case, I 307\u003c\/p\u003e \u003cp\u003e14 Explicit and implicit linear constraints 307\u003c\/p\u003e \u003cp\u003e15 The general linear model, I 310\u003c\/p\u003e \u003cp\u003e16 A singular variance matrix: the general case, II 311\u003c\/p\u003e \u003cp\u003e17 The general linear model, II 314\u003c\/p\u003e \u003cp\u003e18 Generalized least squares 315\u003c\/p\u003e \u003cp\u003e19 Restricted least squares 316\u003c\/p\u003e \u003cp\u003eMiscellaneous exercises\u003ci\u003e \u003c\/i\u003e318\u003c\/p\u003e \u003cp\u003eBibliographical notes\u003ci\u003e \u003c\/i\u003e319\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 Further topics in the linear model 321\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introduction 321\u003c\/p\u003e \u003cp\u003e2 Best quadratic unbiased estimation of σ2 322\u003c\/p\u003e \u003cp\u003e3 The best quadratic and positive unbiased estimator of σ2 322\u003c\/p\u003e \u003cp\u003e4 The best quadratic unbiased estimator of σ2 324\u003c\/p\u003e \u003cp\u003e5 Best quadratic invariant estimation of σ2 326\u003c\/p\u003e \u003cp\u003e6 The best quadratic and positive invariant estimator of σ2 327\u003c\/p\u003e \u003cp\u003e7 The best quadratic invariant estimator of σ2 329\u003c\/p\u003e \u003cp\u003e8 Best quadratic unbiased estimation: multivariate normal case 330\u003c\/p\u003e \u003cp\u003e9 Bounds for the bias of the least-squares estimator of σ\u003csup\u003e2\u003c\/sup\u003e, I 332\u003c\/p\u003e \u003cp\u003e10 Bounds for the bias of the least-squares estimator of σ\u003csup\u003e2\u003c\/sup\u003e, II 333\u003c\/p\u003e \u003cp\u003e11 The prediction of disturbances 335\u003c\/p\u003e \u003cp\u003e12 Best linear unbiased predictors with scalar variance matrix 336\u003c\/p\u003e \u003cp\u003e13 Best linear unbiased predictors with fixed variance matrix, I 338\u003c\/p\u003e \u003cp\u003e14 Best linear unbiased predictors with fixed variance matrix, II 340\u003c\/p\u003e \u003cp\u003e15 Local sensitivity of the posterior mean 341\u003c\/p\u003e \u003cp\u003e16 Local sensitivity of the posterior precision 342\u003c\/p\u003e \u003cp\u003eBibliographical notes\u003ci\u003e \u003c\/i\u003e344\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart Six — Applications to maximum likelihood estimation\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 Maximum likelihood estimation 347\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introduction 347\u003c\/p\u003e \u003cp\u003e2 The method of maximum likelihood (ML) 347\u003c\/p\u003e \u003cp\u003e3 ML estimation of the multivariate normal distribution 348\u003c\/p\u003e \u003cp\u003e4 Symmetry: implicit versus explicit treatment 350\u003c\/p\u003e \u003cp\u003e5 The treatment of positive definiteness 351\u003c\/p\u003e \u003cp\u003e6 The information matrix 352\u003c\/p\u003e \u003cp\u003e7 ML estimation of the multivariate normal distribution: distinct means 354\u003c\/p\u003e \u003cp\u003e8 The multivariate linear regression model 354\u003c\/p\u003e \u003cp\u003e9 The errors-in-variables model 357\u003c\/p\u003e \u003cp\u003e10 The nonlinear regression model with normal errors 359\u003c\/p\u003e \u003cp\u003e11 Special case: functional independence of mean and variance parameters 361\u003c\/p\u003e \u003cp\u003e12 Generalization of Theorem 15.6 362\u003c\/p\u003e \u003cp\u003eMiscellaneous exercises\u003ci\u003e \u003c\/i\u003e364\u003c\/p\u003e \u003cp\u003eBibliographical notes\u003ci\u003e \u003c\/i\u003e365\u003c\/p\u003e \u003cp\u003e\u003cb\u003e16 Simultaneous equations 367\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introduction 367\u003c\/p\u003e \u003cp\u003e2 The simultaneous equations model 367\u003c\/p\u003e \u003cp\u003e3 The identification problem 369\u003c\/p\u003e \u003cp\u003e4 Identification with linear constraints on \u003ci\u003eB \u003c\/i\u003eand Γ only 371\u003c\/p\u003e \u003cp\u003e5 Identification with linear constraints on \u003ci\u003eB\u003c\/i\u003e, Γ, and ∑ 371\u003c\/p\u003e \u003cp\u003e6 Nonlinear constraints 373\u003c\/p\u003e \u003cp\u003e7 FIML: the information matrix (general case) 374\u003c\/p\u003e \u003cp\u003e8 FIML: asymptotic variance matrix (special case) 376\u003c\/p\u003e \u003cp\u003e9 LIML: first-order conditions 378\u003c\/p\u003e \u003cp\u003e10 LIML: information matrix 381\u003c\/p\u003e \u003cp\u003e11 LIML: asymptotic variance matrix 383\u003c\/p\u003e \u003cp\u003eBibliographical notes\u003ci\u003e \u003c\/i\u003e388\u003c\/p\u003e \u003cp\u003e\u003cb\u003e17 Topics in psychometrics 389\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introduction 389\u003c\/p\u003e \u003cp\u003e2 Population principal components 390\u003c\/p\u003e \u003cp\u003e3 Optimality of principal components 391\u003c\/p\u003e \u003cp\u003e4 A related result 392\u003c\/p\u003e \u003cp\u003e5 Sample principal components 393\u003c\/p\u003e \u003cp\u003e6 Optimality of sample principal components 395\u003c\/p\u003e \u003cp\u003e7 One-mode component analysis 395\u003c\/p\u003e \u003cp\u003e8 One-mode component analysis and sample principal components 398\u003c\/p\u003e \u003cp\u003e9 Two-mode component analysis 399\u003c\/p\u003e \u003cp\u003e10 Multimode component analysis 400\u003c\/p\u003e \u003cp\u003e11 Factor analysis 404\u003c\/p\u003e \u003cp\u003e12 A zigzag routine 407\u003c\/p\u003e \u003cp\u003e13 A Newton-Raphson routine 408\u003c\/p\u003e \u003cp\u003e14 Kaiser’s varimax method 412\u003c\/p\u003e \u003cp\u003e15 Canonical correlations and variates in the population 414\u003c\/p\u003e \u003cp\u003e16 Correspondence analysis 417\u003c\/p\u003e \u003cp\u003e17 Linear discriminant analysis 418\u003c\/p\u003e \u003cp\u003eBibliographical notes\u003ci\u003e \u003c\/i\u003e419\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart Seven — Summary\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e18 Matrix calculus: the essentials 423\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introduction 423\u003c\/p\u003e \u003cp\u003e2 Differentials 424\u003c\/p\u003e \u003cp\u003e3 Vector calculus 426\u003c\/p\u003e \u003cp\u003e4 Optimization 429\u003c\/p\u003e \u003cp\u003e5 Least squares 431\u003c\/p\u003e \u003cp\u003e6 Matrix calculus 432\u003c\/p\u003e \u003cp\u003e7 Interlude on linear and quadratic forms 434\u003c\/p\u003e \u003cp\u003e8 The second differential 434\u003c\/p\u003e \u003cp\u003e9 Chain rule for second differentials 436\u003c\/p\u003e \u003cp\u003e10 Four examples 438\u003c\/p\u003e \u003cp\u003e11 The Kronecker product and vec operator 439\u003c\/p\u003e \u003cp\u003e12 Identification 441\u003c\/p\u003e \u003cp\u003e13 The commutation matrix 442\u003c\/p\u003e \u003cp\u003e14 From second differential to Hessian 443\u003c\/p\u003e \u003cp\u003e15 Symmetry and the duplication matrix 444\u003c\/p\u003e \u003cp\u003e16 Maximum likelihood 445\u003c\/p\u003e \u003cp\u003eFurther reading 448\u003c\/p\u003e \u003cp\u003eBibliography 449\u003c\/p\u003e \u003cp\u003eIndex of symbols 467\u003c\/p\u003e \u003cp\u003eSubject index 471\u003c\/p\u003e  \u003cp\u003e\u003cb\u003eJAN R. MAGNUS\u003c\/b\u003e is Emeritus Professor at the Department of Econometrics \u0026amp; Operations Research, Tilburg University, and Extraordinary Professor at the Department of Econometrics \u0026amp; Operations Research, Vrije University, Amsterdam. He is research fellow of CentER and the Tinbergen Institute. He has co-authored nine books and is the author of over 100 scientific papers. \u003c\/p\u003e\u003cp\u003e\u003cb\u003eHEINZ NEUDECKER\u003c\/b\u003e (1933-2017) was Professor of Econometrics at the University of Amsterdam from 1972 until his retirement in 1998.    \u003c\/p\u003e\u003cp\u003e\u003cb\u003eA BRAND NEW, FULLY UPDATED EDITION OF A POPULAR CLASSIC ON MATRIX DIFFERENTIAL CALCULUS WITH APPLICATIONS IN STATISTICS AND ECONOMETRICS\u003c\/b\u003e \u003c\/p\u003e\u003cp\u003eThis exhaustive, self-contained book on matrix theory and matrix differential calculus provides a treatment of matrix calculus based on differentials and shows how easy it is to use this theory once you have mastered the technique. Jan Magnus, who, along with the late Heinz Neudecker, pioneered the theory, develops it further in this new edition and provides many examples along the way to support it. \u003c\/p\u003e\u003cp\u003eMatrix calculus has become an essential tool for quantitative methods in a large number of applications, ranging from social and behavioral sciences to econometrics. It is still relevant and used today in a wide range of subjects such as the biosciences and psychology. \u003ci\u003eMatrix Differential Calculus with Applications in Statistics and Econometrics, Third Edition\u003c\/i\u003e contains all of the essentials of multivariable calculus with an emphasis on the use of differentials. It starts by presenting a concise, yet thorough overview of matrix algebra, then goes on to develop the theory of differentials. The rest of the text combines the theory and application of matrix differential calculus, providing the practitioner and researcher with both a quick review and a detailed reference. \u003c\/p\u003e\u003cul\u003e \u003cli\u003eFulfills the need for an updated and unified treatment of matrix differential calculus\u003c\/li\u003e \u003cli\u003eContains many new examples and exercises based on questions asked of the author over the years\u003c\/li\u003e \u003cli\u003eCovers new developments in field and features new applications\u003c\/li\u003e \u003cli\u003eWritten by a leading expert and pioneer of the theory\u003c\/li\u003e \u003cli\u003ePart of the Wiley Series in Probability and Statistics\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003e\u003ci\u003eMatrix Differential Calculus With Applications in Statistics and Econometrics, Third Edition\u003c\/i\u003e is an ideal text for graduate students and academics studying the subject, as well as for postgraduates and specialists working in biosciences and psychology.\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47989589770469,"sku":"NP9781119541202","price":116.95,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9781119541202.jpg?v=1761784717","url":"https:\/\/k12savings.com\/es\/products\/matrix-differential-calculus-with-applications-in-statistics-and-econometrics-isbn-9781119541202","provider":"K12savings","version":"1.0","type":"link"}