This book represents the first synthesis of the considerable body of new
research into positive definite matrices. These matrices play the same
role in noncommutative analysis as positive real numbers do in classical
analysis. They have theoretical and computational uses across a broad
spectrum of disciplines, including calculus, electrical engineering,
statistics, physics, numerical analysis, quantum information theory, and
geometry. Through detailed explanations and an authoritative and
inspiring writing style, Rajendra Bhatia carefully develops general
techniques that have wide applications in the study of such matrices.
Bhatia introduces several key topics in functional analysis, operator
theory, harmonic analysis, and differential geometry--all built around
the central theme of positive definite matrices. He discusses positive
and completely positive linear maps, and presents major theorems with
simple and direct proofs. He examines matrix means and their
applications, and shows how to use positive definite functions to derive
operator inequalities that he and others proved in recent years. He
guides the reader through the differential geometry of the manifold of
positive definite matrices, and explains recent work on the geometric
mean of several matrices.
Positive Definite Matrices is an informative and useful reference book
for mathematicians and other researchers and practitioners. The numerous
exercises and notes at the end of each chapter also make it the ideal
textbook for graduate-level courses.