Positive Definite Matrices

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.

"There is no obvious competitor for Bhatia's book, due in part to its focus, but also because it contains some very recent material drawn from research articles. Beautifully written and intelligently organised, Positive Definite Matrices is a welcome addition to the literature. Readers who admired his Matrix Analysis will no doubt appreciate this latest book of Rajendra Bhatia."---Douglas Farenick, Image

"Written by an expert in the area, the book presents in an accessible manner a lot of important results from the realm of positive matrices and of their applications. . . . The book can be used for graduate courses in linear algebra, or as supplementary material for courses in operator theory, and as a reference book by engineers and researchers working in the applied field of quantum information."---S. Cobzas, Studia Universitatis Babes-Bolyai, Mathematica

"This is a monograph for mathematicians interested in an important realm of matrix-analytic ideas. Like the author's distinguished book, Matrix Analysis, it will be a convenient and much-quoted reference source. There are many wonderful insights in a first-rate exposition of important ideas not easily extracted from other sources. The scholarship is impeccable."—Roger A. Horn, University of Utah

"I believe that every expert in matrix analysis can find something new in this book. Bhatia presents some important material in several topics related to positive definite matrices including positive linear maps, completely positive maps, matrix means, positive definite functions, and geometry of positive definite matrices. There are many beautiful results, useful techniques, and ingenious ideas here. Bhatia's writing style has always been concise, clear, and illuminating."—Xingzhi Zhan, East China Normal University