This book represents the first synthesis of the considerable body of new research in positive definite matrices. Through detailed explanations and an authoritative and inspiring writing style, R. Bhatia carefully develops general techniques that have wide applications in the study of such matrices. The book begins with a quick review of some of the basic properties of positive matrices. The author introduces several key topics in functional analysis, operator theory, harmonic analysis and differential geometry, all built around the central theme of positive definite matrices. Chapters 2 and 3 are devoted to a study of positive and completely positive maps and, in particular, on their use in proving inequalities. In chapter 4 the author discusses means of two positive definite matrices with special emphasis on the geometric mean. Among some spectacular applications of these ideas, the author includes proofs of some theorems in the field of matrix convex functions and two of the most famous theorems in the field of quantum mechanical entropy. Chapter 5 gives a quick introduction to positive definite functions on the real line. Again, special attention is given to various means of matrices. Many of these results come from recent research work. Chapter 6 presents some standard and important theorems of Riemannian geometry as seen from the perspective of matrix analysis. Notes and references are appended to each chapter.

The textbook is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes also make it ideal for graduate-level courses.

Reviewer:

kn