## 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.
"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." "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, "This is an outstanding book. Its exposition is both concise and leisurely at the same time."
"This is a monograph for mathematicians interested in an important realm of matrix-analytic ideas. Like the author's distinguished book, "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."
- Princeton Series in Applied Mathematics
Ingrid Daubechies, Weinan E, Jan Karel Lenstra, and Endre Süli, Editors
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