This book synthesizes extensive new research on positive definite matrices, which are crucial in noncommutative analysis, akin to the role of positive real numbers in classical analysis. These matrices have theoretical and computational applications across various fields, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. With detailed explanations and an authoritative writing style, the author develops general techniques applicable to the study of these matrices. Key topics in functional analysis, operator theory, harmonic analysis, and differential geometry are introduced, all centered on positive definite matrices. The author discusses positive and completely positive linear maps, presenting major theorems with straightforward proofs. The book explores matrix means and their applications, demonstrating how positive definite functions can be used to derive operator inequalities established in recent years. Additionally, the author guides readers through the differential geometry of the manifold of positive definite matrices and recent findings on the geometric mean of multiple matrices. This work serves as an informative reference for mathematicians and researchers, while the exercises and notes at the end of each chapter make it an ideal textbook for graduate-level courses.
