Matrix differential calculus with applications in statistics and econometrics. (English) Zbl 0651.15001

Magnus, Jan R.; Neudecker, Heinz

Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics. Chichester etc.: John Wiley & Sons. XVII, 393 p.; £24.50 (1988).

Matrix derivatives arise in practice when dealing with certain problems in statistics and econometrics in the areas of multivariate linear regression and maximum likelihood estimation. This book is a self- contained, detailed and coherent account of matrix differential calculus, together with a considerable number of significant applications. What makes it different from other books on the subject is its emphasis on the matrix differential rather than the derivatives. This is not just an idiosyncracy on the part of the authors. For they show, time and again, that this approach leads to an elegance and clarity of notation in a subject where complicated expressions abound.
The book contains 17 chapters divided into 6 parts. Chapter headings give a very good idea of the contents of the book. Part One - Matrices. 1. Basic properties of vectors and matrices. 2. Kronecker products, the vec operator and the Moore-Penrose inverse. 3. Miscellaneous matrix results. Part Two - Differentials: the theory. 4. Mathematical preliminaries. 5. Differentials and differentiability. 6. The second differential. 7. Static optimization. Part Three - Differentials: the practice. 8. Some important differentials. 9. First-order differentials and Jacobian matrices. 10. Second order differentials and Hessian matrices. Part Four - Inequalities. 11. Inequalities. Part Five - The linear model. 12. Statistical preliminaries. 13. The linear regression model. 14. Further topics in the linear model. Part Six - Applications to maximum likelihood estimation. 15. Maximum likelihood estimation. 16. Simultaneous equations. 17. Topics in psychometrics. There are exercises of various degrees of difficulty at the end of each section.
All in all this is an excellent book which serves not only as a text but as a good source of reference. (The reviewer’s criticisms are only of a very minor nature. Theorem 29 is needed to prove Theorem 28. In the proof of the Bolzano-Weierstrass theorem it is not explained why the nested intervals intersect in a point. The notation (A:B) partitioning a matrix into two blocks of columns is used but not defined. In chapter 13 the words “regressor” and “non-stochastic” are not defined.)

Reviewer: F.J.Gaines

MSC:

15-02	Research exposition (monographs, survey articles) pertaining to linear algebra
62J05	Linear regression; mixed models
62P15	Applications of statistics to psychology
53A45	Differential geometric aspects in vector and tensor analysis
26B12	Calculus of vector functions
62H25	Factor analysis and principal components; correspondence analysis
15A09	Theory of matrix inversion and generalized inverses
15A45	Miscellaneous inequalities involving matrices
15A42	Inequalities involving eigenvalues and eigenvectors

Keywords:

multivariate linear regression; maximum likelihood estimation; matrix differential calculus; vectors; matrices; Kronecker products; vec operator; Moore-Penrose inverse; Static optimization; Jacobian matrices; Hessian matrices; Inequalities; psychometrics