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.

Darrell A. Turkington is a Professor of Economics at the University of Western Australia. His numerous publications include articles in leading international journals such as the Journal of the American Statistical Association, the International Economic Review and the Journal of Econometrics. He is also the author of Instrumental Variables (Cambridge University Press, 1985, with Roger J. Bowden), Matrix Calculus and Zero-One Matrices: Statistical and Econometric Applications (Cambridge University Press, 2002) and Mathematical Tools for Economics (2007). Professor Turkington received his Ph.D. in theoretical economics from the University of California, Berkeley.