Matrix Differential Calculus with Applications in Statistics and Econometrics.

Cover -- Title Page -- Copyright -- Contents -- Preface -- Part One - Matrices -- Chapter 1 Basic properties of vectors and matrices -- 1 Introduction -- 2 Sets -- 3 Matrices: addition and multiplication -- 4 The transpose of a matrix -- 5 Square matrices -- 6 Linear forms and quadratic forms -- 7 The rank of a matrix -- 8 The inverse -- 9 The determinant -- 10 The trace -- 11 Partitioned matrices -- 12 Complex matrices -- 13 Eigenvalues and eigenvectors -- 14 Schur's decomposition theorem -- 15 The Jordan decomposition -- 16 The singular-value decomposition -- 17 Further results concerning eigenvalues -- 18 Positive (semi)definite matrices -- 19 Three further results for positive definite matrices -- 20 A useful result -- 21 Symmetric matrix functions -- Miscellaneous exercises -- Bibliographical notes -- Chapter 2 Kronecker products, vec operator, and Moore-Penrose inverse -- 1 Introduction -- 2 The Kronecker product -- 3 Eigenvalues of a Kronecker product -- 4 The vec operator -- 5 The Moore-Penrose (MP) inverse -- 6 Existence and uniqueness of the MP inverse -- 7 Some properties of the MP inverse -- 8 Further properties -- 9 The solution of linear equation systems -- Miscellaneous exercises -- Bibliographical notes -- Chapter 3 Miscellaneous matrix results -- 1 Introduction -- 2 The adjoint matrix -- 3 Proof of Theorem 3.1 -- 4 Bordered determinants -- 5 The matrix equation AX = 0 -- 6 The Hadamard product -- 7 The commutation matrix Kmn -- 8 The duplication matrix Dn -- 9 Relationship between Dn+1 and Dn, I -- 10 Relationship between Dn+1 and Dn, II -- 11 Conditions for a quadratic form to be positive (negative) subject to linear constraints -- 12 Necessary and sufficient conditions for r(A : B) = r(A) + r(B) -- 13 The bordered Gramian matrix -- 14 The equations X1A + X2B′ = G1,X1B = G2 -- Miscellaneous exercises -- Bibliographical notes.

Part Two - Differentials: the theory -- Chapter 4 Mathematical preliminaries -- 1 Introduction -- 2 Interior points and accumulation points -- 3 Open and closed sets -- 4 The Bolzano-Weierstrass theorem -- 5 Functions -- 6 The limit of a function -- 7 Continuous functions and compactness -- 8 Convex sets -- 9 Convex and concave functions -- Bibliographical notes -- Chapter 5 Differentials and differentiability -- 1 Introduction -- 2 Continuity -- 3 Differentiability and linear approximation -- 4 The differential of a vector function -- 5 Uniqueness of the differential -- 6 Continuity of differentiable functions -- 7 Partial derivatives -- 8 The first identification theorem -- 9 Existence of the differential, I -- 10 Existence of the differential, II -- 11 Continuous differentiability -- 12 The chain rule -- 13 Cauchy invariance -- 14 The mean-value theorem for real-valued functions -- 15 Differentiable matrix functions -- 16 Some remarks on notation -- 17 Complex differentiation -- Miscellaneous exercises -- Bibliographical notes -- Chapter 6 The second differential -- 1 Introduction -- 2 Second-order partial derivatives -- 3 The Hessian matrix -- 4 Twice differentiability and second-order approximation, I -- 5 Definition of twice differentiability -- 6 The second differential -- 7 Symmetry of the Hessian matrix -- 8 The second identification theorem -- 9 Twice differentiability and second-order approximation, II -- 10 Chain rule for Hessian matrices -- 11 The analog for second differentials -- 12 Taylor's theorem for real-valued functions -- 13 Higher-order differentials -- 14 Real analytic functions -- 15 Twice differentiable matrix functions -- Bibliographical notes -- Chapter 7 Static optimization -- 1 Introduction -- 2 Unconstrained optimization -- 3 The existence of absolute extrema -- 4 Necessary conditions for a local minimum.

5 Sufficient conditions for a local minimum: first-derivative test -- 6 Sufficient conditions for a local minimum: second-derivative test -- 7 Characterization of differentiable convex functions -- 8 Characterization of twice differentiable convex functions -- 9 Sufficient conditions for an absolute minimum -- 10 Monotonic transformations -- 11 Optimization subject to constraints -- 12 Necessary conditions for a local minimum under constraints -- 13 Sufficient conditions for a local minimum under constraints -- 14 Sufficient conditions for an absolute minimum under constraints -- 15 A note on constraints in matrix form -- 16 Economic interpretation of Lagrange multipliers -- Appendix: the implicit function theorem -- Bibliographical notes -- Part Three - Differentials: the practice -- Chapter 8 Some important differentials -- 1 Introduction -- 2 Fundamental rules of differential calculus -- 3 The differential of a determinant -- 4 The differential of an inverse -- 5 Differential of the Moore-Penrose inverse -- 6 The differential of the adjoint matrix -- 7 On differentiating eigenvalues and eigenvectors -- 8 The continuity of eigenprojections -- 9 The differential of eigenvalues and eigenvectors: symmetric case -- 10 Two alternative expressions for dλ -- 11 Second differential of the eigenvalue function -- Miscellaneous exercises -- Bibliographical notes -- Chapter 9 First-order differentials and Jacobian matrices -- 1 Introduction -- 2 Classification -- 3 Derisatives -- 4 Derivatives -- 5 Identification of Jacobian matrices -- 6 The first identification table -- 7 Partitioning of the derivative -- 8 Scalar functions of a scalar -- 9 Scalar functions of a vector -- 10 Scalar functions of a matrix, I: trace -- 11 Scalar functions of a matrix, II: determinant -- 12 Scalar functions of a matrix, III: eigenvalue -- 13 Two examples of vector functions.

14 Matrix functions -- 15 Kronecker products -- 16 Some other problems -- 17 Jacobians of transformations -- Bibliographical notes -- Chapter 10 Second-order differentials and Hessian matrices -- 1 Introduction -- 2 The second identification table -- 3 Linear and quadratic forms -- 4 A useful theorem -- 5 The determinant function -- 6 The eigenvalue function -- 7 Other examples -- 8 Composite functions -- 9 The eigenvector function -- 10 Hessian of matrix functions, I -- 11 Hessian of matrix functions, II -- Miscellaneous exercises -- Part Four - Inequalities -- Chapter 11 Inequalities -- 1 Introduction -- 2 The Cauchy-Schwarz inequality -- 3 Matrix analogs of the Cauchy-Schwarz inequality -- 4 The theorem of the arithmetic and geometric means -- 5 The Rayleigh quotient -- 6 Concavity of λ1 and convexity of λn -- 7 Variational description of eigenvalues -- 8 Fischer's min-max theorem -- 9 Monotonicity of the eigenvalues -- 10 The Poincar´e separation theorem -- 11 Two corollaries of Poincar´e's theorem -- 12 Further consequences of the Poincar´e theorem -- 13 Multiplicative version -- 14 The maximum of a bilinear form -- 15 Hadamard's inequality -- 16 An interlude: Karamata's inequality -- 17 Karamata's inequality and eigenvalues -- 18 An inequality concerning positive semidefinite matrices -- 19 A representation theorem for (Σapi)1/p -- 20 A representation theorem for (trAp)1/p -- 21 H¨older's inequality -- 22 Concavity of log |A| -- 23 Minkowski's inequality -- 24 Quasilinear representation of |A|1/n -- 25 Minkowski's determinant theorem -- 26 Weighted means of order p -- 27 Schl¨omilch's inequality -- 28 Curvature properties of Mp(x, a) -- 29 Least squares -- 30 Generalized least squares -- 31 Restricted least squares -- 32 Restricted least squares: matrix version -- Miscellaneous exercises -- Bibliographical notes -- Part Five - The linear model.

Chapter 12 Statistical preliminaries -- 1 Introduction -- 2 The cumulative distribution function -- 3 The joint density function -- 4 Expectations -- 5 Variance and covariance -- 6 Independence of two random variables -- 7 Independence of n random variables -- 8 Sampling -- 9 The one-dimensional normal distribution -- 10 The multivariate normal distribution -- 11 Estimation -- Miscellaneous exercises -- Bibliographical notes -- Chapter 13 The linear regression model -- 1 Introduction -- 2 Affine minimum-trace unbiased estimation -- 3 The Gauss-Markov theorem -- 4 The method of least squares -- 5 Aitken's theorem -- 6 Multicollinearity -- 7 Estimable functions -- 8 Linear constraints: the case M(R′) ⊂M(X′) -- 9 Linear constraints: the general case -- 10 Linear constraints: the case M(R′) ∩M(X′) = {0} -- 11 A singular variance matrix: the case M(X) ⊂M(V ) -- 12 A singular variance matrix: the case r(X′V +X) = r(X) -- 13 A singular variance matrix: the general case, I -- 14 Explicit and implicit linear constraints -- 15 The general linear model, I -- 16 A singular variance matrix: the general case, II -- 17 The general linear model, II -- 18 Generalized least squares -- 19 Restricted least squares -- Miscellaneous exercises -- Bibliographical notes -- Chapter 14 Further topics in the linear model -- 1 Introduction -- 2 Best quadratic unbiased estimation of σ2 -- 3 The best quadratic and positive unbiased estimator of σ2 -- 4 The best quadratic unbiased estimator of σ2 -- 5 Best quadratic invariant estimation of σ2 -- 6 The best quadratic and positive invariant estimator of σ2 -- 7 The best quadratic invariant estimator of σ2 -- 8 Best quadratic unbiased estimation: multivariate normal case -- 9 Bounds for the bias of the least-squares estimator of σ2, I -- 10 Bounds for the bias of the least-squares estimator of σ2, II.

11 The prediction of disturbances.

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