Matrix Differential Calculus with Applications in Statistics and Econometrics

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.

Autorentext

JAN R. MAGNUS is Emeritus Professor at the Department of Econometrics & Operations Research, Tilburg University, and Extraordinary Professor at the Department of Econometrics & 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. HEINZ NEUDECKER (1933-2017) was Professor of Econometrics at the University of Amsterdam from 1972 until his retirement in 1998.

Klappentext
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.

Inhalt
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...