Matrix analysis. (English) Zbl 0863.15001

Bhatia, Rajendra

Graduate Texts in Mathematics. 169. New York, NY: Springer. xi, 347 p. (1996).

The book under review is devoted to matrix analysis in the spirit of functional analysis and with great emphasis on the art of deriving matrix inequalities. This art can be compared to that of cutting diamonds: it requires hard tools and a delicate use of them.
The text consists of ten chapters: The first chapter establishes notations and introduces preliminaries from linear and multilinear algebra. Special attention is given to tensor products and symmetry classes. In the next three chapters well elaborated background material is explained which should be included in any course on matrix analysis. Chapter 5 on operator monotone and operator convex functions presents more advanced and more special material. Chapters 6 and 8 are devoted to perturbation of spectra, a topic of much importance in numerical analysis, physics and engineering. Chapter 9 (selection of matrix inequalities) and Chapter 10 (Perturbation of matrix functions) also have been of broad interest in several areas.
The presentation of the material covered in the book is very clear. The contents of each chapter are briefly summarized in its first paragraph, while at the end of each chapter detailed references and comments to the original published sources and the most important related papers are collected. These Notes and References give insight, explain the ideas behind many of the concepts, and point out connections. Several exercises are scattered in the text and in the Problem section in each chapter. These exercises range in difficulty from the “quite easy” to hard enough to yield the contents of research papers.
On the whole, the author has managed to create a highly readable and attractive account of the subject. The book is a must for anyone working in matrix analysis; it can be recommended to graduate students as well as to specialists.

Reviewer: J.Szilasi (Debrecen)

MSC:

15-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra
15A60	Norms of matrices, numerical range, applications of functional analysis to matrix theory
15B51	Stochastic matrices
15A42	Inequalities involving eigenvalues and eigenvectors
49R50	Variational methods for eigenvalues of operators (MSC2000)
15A45	Miscellaneous inequalities involving matrices
47A55	Perturbation theory of linear operators
65F15	Numerical computation of eigenvalues and eigenvectors of matrices

Keywords:

textbook; matrix analysis; tensor products; symmetry classes; operator convex functions; matrix inequalities; matrix functions; exercises