Generalized Vectorization, Cross-Products, and Matrix Calculus

This book presents the reader with new operators and matrices that arise in the area of matrix calculus. The properties of these mathematical concepts are investigated and linked with zero-one matrices such as the commutation matrix. Elimination and duplication matrices are revisited and partitioned into submatrices. Studying the properties of these submatrices facilitates achieving new results for the original matrices themselves. Different concepts of matrix derivatives are presented and transformation principles linking these concepts are obtained. One of these concepts is used to derive new matrix calculus results, some involving the new operators and others the derivatives of the operators themselves. The last chapter contains applications of matrix calculus, including optimization, differentiation of log-likelihood functions, iterative interpretations of maximum likelihood estimators and a Lagrangian multiplier test for endogeneity. Advance praise: 'A very neat treatment of matrix calculus. There is no doubt that the new operators and matrices presented in the book will see their applications in many areas of econometrics.' Yong Bao, Purdue University Advance praise: 'This book is very clearly written in a text style that conveys what needs to be said with no superfluous discussion. It represents a substantial contribution to our understanding of a difficult area. It is a beautiful book, and destined to become a classic.' Ross Maller, Australian National University1. Mathematical prerequisites; 2. Zero-one matrices; 3. Elimination and duplication matrices; 4. Matrix calculus; 5. New matrix calculus results; 6. Applications.