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A brand new, fully updated edition of a popular classic on matrix differential calculus with applications in statistics and econometrics

This 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.

Matrix 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. Matrix Differential Calculus with Applications in Statistics and Econometrics, Third Edition 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.

Fulfills the need for an updated and unified treatment of matrix differential calculus
Contains many new examples and exercises based on questions asked of the author over the years
Covers new developments in field and features new applications
Written by a leading expert and pioneer of the theory
Part of the Wiley Series in Probability and Statistics

Matrix Differential Calculus With Applications in Statistics and Econometrics Third Edition is an ideal text for graduate students and academics studying the subject, as well as for postgraduates and specialists working in biosciences and psychology.




Table of contents

Preface xiii

Part One - Matrices

1 Basic properties of vectors and matrices 3

1 Introduction 3

2 Sets 3

3 Matrices: addition and multiplication 4

4 The transpose of a matrix 6

5 Square matrices 6

6 Linear forms and quadratic forms 7

7 The rank of a matrix 9

8 The inverse 10

9 The determinant 10

10 The trace 11

11 Partitioned matrices 12

12 Complex matrices 14

13 Eigenvalues and eigenvectors 14

14 Schur's decomposition theorem 17

15 The Jordan decomposition 18

16 The singular-value decomposition 20

17 Further results concerning eigenvalues 20

18 Positive (semi)definite matrices 23

19 Three further results for positive definite matrices 25

20 A useful result 26

21 Symmetric matrix functions 27

Miscellaneous exercises 28

Bibliographical notes 30

2 Kronecker products, vec operator, and Moore-Penrose inverse 31

1 Introduction 31

2 The Kronecker product 31

3 Eigenvalues of a Kronecker product 33

4 The vec operator 34

5 The Moore-Penrose (MP) inverse 36

6 Existence and uniqueness of the MP inverse 37

7 Some properties of the MP inverse 38

8 Further properties 39

9 The solution of linear equation systems 41

Miscellaneous exercises 43

Bibliographical notes 45

3 Miscellaneous matrix results 47

1 Introduction 47

2 The adjoint matrix 47

3 Proof of Theorem 3.1 49

4 Bordered determinants 51

5 The matrix equation AX = 0 51

6 The Hadamard product 52

7 The commutation matrix Kmn 54

8 The duplication matrix Dn 56

9 Relationship between Dn+1 and Dn, I 58

10 Relationship between Dn+1 and Dn, II 59

11 Conditions for a quadratic form to be positive (negative) subject to linear constraints 60

12 Necessary and sufficient conditions for r(A : B) = r(A) + r(B) 63

13 The bordered Gramian matrix 65

14 The equations X1A + X2B´ = G1,X1B = G2 67

Miscellaneous exercises 69

Bibliographical notes 70

Part Two - Differentials: the theory

4 Mathematical preliminaries 73

1 Introduction 73

2 Interior points and accumulation points 73

3 Open and closed sets 75

4 The Bolzano-Weierstrass theorem 77

5 Functions 78

6 The limit of a function 79

7 Continuous functions and compactness 80

8 Convex sets 81

9 Convex and concave functions 83

Bibliographical notes 86

5 Differentials and differentiability 87

1 Introduction 87

2 Continuity 88

3 Differentiability and linear approximation 90

4 The differential of a vector function 91

5 Uniqueness of the differential 93

6 Continuity of differentiable functions 94

7 Partial derivatives 95

8 The first identification theorem 96

9 Existence of the differential, I 97

10 Existence of the differential, II 99

11 Continuous differentiability 100

12 The chain rule 100

13 Cauchy invariance 102

14 The mean-value theorem for real-valued functions 103

15 Differentiable matrix functions 104

16 Some remarks on notation 106

17 Complex differentiation 108

Miscellaneous exercises 110

Bibliographical notes 110

6 The second differential 111

1 Introduction 111

2 Second-order partial derivatives 111

3 The Hessian matrix 112

4 Twice differentiability and second-order approximation, I 113

5 Definition of twice differentiability 114

6 The second differential 115

7 Symmetry of the Hessian matrix 117

8 The second identification theorem 119

9 Twice differentiability and second-order approximation, II 119

10 Chain rule for Hessian matrices 121

11 The analog for second differentials 123

12 Taylor's theorem for real-valued functions 124

13 Higher-order differentials 125

14 Real analytic functions 125

15 Twice differentiable matrix functions 126

Bibliographical notes 127

7 Static optimization 129

1 Introduction 129

2 Unconstrained optimization 130

3 The existence of absolute extrema 131

4 Necessary conditions for a local minimum 132

5 Sufficient conditions for a local minimum: first-derivative test 134

6 Sufficient conditions for a local minimum: second-derivative test 136

7 Characterization of differentiable convex functions 138

8 Characterization of twice differentiable convex functions 141

9 Sufficient conditions for an absolute minimum 142

10 Monotonic transformations 143

11 Optimization subject to constraints 144

12 Necessary conditions for a local minimum under constraints 145

13 Sufficient conditions for a local minimum under constraints 149

14 Sufficient conditions for an absolute minimum under constraints 154

15 A note on constraints in matrix form 155

16 Economic interpretation of Lagrange multipliers 155

Appendix: the implicit function theorem 157

Bibliographical notes 159

Part Three - Differentials: the practice

8 Some important differentials 163

1 Introduction 163

2 Fundamental rules of differential calculus 163

3 The differential of a determinant 165

4 The dif