Kronecker Products and Matrix Calculus: With Applications (Mathematics and Its Applications)

Table of Contents......Page 5
Author's Preface......Page 7
Symbols and Notation Used......Page 9
1.2 UNIT VECTORS AND ELEMENTARY MATRICES......Page 11
1.3 DECOMPOSITIONS OF A MATRIX......Page 13
1.4 THE TRACE FUNCTION......Page 17
1.5 THE VEC OPERATOR......Page 18
Problems for Chapter 1......Page 20
2.2 DEFINITION OF THE KRONECKER PRODUCT......Page 21
2.3 SOME PROPERTIES AND RULES FOR KRONECKER PRODUCTS......Page 23
2.4 DEFINITION OF THE KRONECKER SUM......Page 30
2.5 THE PERMUTATION MATRIX ASSOCIATING vec X AND vec X'......Page 32
Problems of Chapter 2......Page 35
3.2 THE DERIVATIVE OF A MATRIX......Page 37
3.3 PROBLEM 1......Page 38
3.4 PROBLEM 2......Page 40
3.5 PROBLEM 3......Page 41
3.6 PROBLEM 4......Page 42
3.7 PROBLEM 5......Page 44
3.8 PROBLEM 6......Page 45
4.1 INTRODUCTION......Page 51
4.2 THE DERIVATIVES OF VECTORS......Page 52
4.3 THE CHAIN RULE FOR VECTORS......Page 54
4.4 THE DERIVATIVE OF SCALAR FUNCTIONS OF A MATRIX WITHRESPECT TO THE MATRIX......Page 56
4,5 THE DERIVATIVE OF A MATRIX WITH RESPECT TO ONE OF ITSELEMENTS AND CONVERSELY......Page 60
4.6 THE DERIVATIVES OF THE POWERS OF A MATRIX......Page 67
Problems for Chapter 4......Page 68
5.2 DERIVATIVES OF MATRICES AND KRONECKER PRODUCTS......Page 70
5.3 THE DETERMINATION OF (a vec X)/(3 vec Y) FOR MORECOMPLICATED EQUATIONS......Page 72
5.4 MORE ON DERIVATIVES OF SCALAR FUNCTIONS WITHRESPECT TO A MATRIX......Page 75
5.5 THE MATRIX DIFFERENTIAL......Page 78
Problems for Chapter 5......Page 80
6.2 THE DEFINITIONS AND SOME RESULTS......Page 81
'6.3 PRODUCT RULES FOR MATRICES......Page 84
6.4 THE CHAIN RULE FOR THE DERIVATIVE OF A MATRIX WITHRESPECT TO A MATRIX......Page 88
Problems for Chapter 6......Page 92
7.2 THE PROBLEMS OF LEAST SQUARES AND CONSTRAINEDOPTIMISATION IN SCALAR VARIABLES......Page 94
7.3 PROBLEM 1 - MATRIX CALCULUS APPROACH TO THE PROBLEMSOF LEAST SQUARES AND CONSTRAINED OPTIMISATION......Page 96
7.4 PROBLEM 2 - THE GENERAL LEAST SQUARES PROBLEM......Page 100
7.5 PROBLEM 3 - MAXIMUM LIKELIHOOD ESTIMATE OF THEMULTIVARIATE NORMAL......Page 102
7.6 PROBLEM 4 - EVALUATION OF THE JACOBIANS OF SOMETRANSFORMATIONS......Page 104
7.7 PROBLEM 5 - TO FIND THE DERIVATIVE OF AN EXPONENTIALMATRIX WITH RESPECT TO A MATRIX......Page 108
CHAPTER I......Page 111
CHAPTER 2......Page 112
CHAPTER 4......Page 114
CHAPTER 5......Page 116
CHAPTER 6......Page 119
Tables of Formulae andDerivatives......Page 121
Bibliography......Page 126
Index......Page 129