Matrix Differential Calculus with Applications in Statistics and Econometrics

Article excerpt

Jan R. Magnus and Heinz Neudecker. Chichester, U.K.: John Wiley, 1988. xvii + 393 pp. $51.95.

This book delivers far more than its rather misleading title promises. Statisticians and econometricians who deal with matrix theory in their everyday research will be extremely pleased to discover such a complete textbook (written especially for them). It fills a long-felt need for an exhaustive, unified, and self-contained treatment of matrix theory and matrix differential calculus.

While reading this book, I continued to be pleasantly surprised by the variety and usefulness of its contents. It covers an outstanding breadth of topics: the topology of [R.sup.n], inequalities (including matrix inequalities), the linear model (in a modern presentation), the multivariate normal model, maximum likelihood estimation, the errors-in-variables model, nonlinear models, principal components, factor analysis, canonical correlations, optimization, and more.

The book's approach to matrix differential calculus, based on differentials rather than derivatives, is its main novelty. Part I is excellent. In three chapters, both the basic and more advanced aspects of matrix algebra are covered, often with interesting proofs for well-known and more recent results. This part makes a much better reference for matrix algebra than the discussions contained in most multivariate analysis text-books. Part II develops the theory of differentials. It includes helpful rules for identifying Jacobian and Hessian matrices, and a chapter on optimization via differentials. Part III contains the rules for using differentials and lists differentials of the most common functions of vectors and matrices. Part IV, consisting of a single chapter about inequalities and their extension to matrix inequalities, is especially interesting. …