![]() ![]() Sometimes this condition can be confirmed from the definition of. The definition requires the positivity of the quadratic form. For example, the matrixĪ sufficient condition for a symmetric matrix to be positive definite is that it has positive diagonal elements and is diagonally dominant, that is, for all. Satisfying these inequalities is not sufficient for positive definiteness. Bhatia's writing style has always been concise, clear, and illuminating.A real matrix is symmetric positive definite if it is symmetric ( is equal to its transpose, ) andīy making particular choices of in this definition we can derive the inequalities There are many beautiful results, useful techniques, and ingenious ideas here. 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. "I believe that every expert in matrix analysis can find something new in this book. There are many wonderful insights in a first-rate exposition of important ideas not easily extracted from other sources. Like the author's distinguished book, Matrix Analysis, it will be a convenient and much-quoted reference source. "This is a monograph for mathematicians interested in an important realm of matrix-analytic ideas. Its exposition is both concise and leisurely at the same time." -Jaspal Singh Aujla, Zentralblatt MATH Readers who admired his Matrix Analysis will no doubt appreciate this latest book of Rajendra Bhatia." -Douglas Farenick, Image Beautifully written and intelligently organised, Positive Definite Matrices is a welcome addition to the literature. "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. Cobzas, Studia Universitatis Babes-Bolyai, Mathematica ![]() 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. "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 numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. 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. 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 discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. 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.īhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry - all built around the central theme of positive definite matrices. 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. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. ![]() This book represents the first synthesis of the considerable body of new research into positive definite matrices. ![]()
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